#505704

order	512
nilpotency class	4
isoclinism	isoclinic class of #500850
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27662
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

461 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,12,6,13,7,26)(3,14,27,41,16,40,19,49)(4,20,28,51,22,52,23,54)(5,24,29,37,25,55,15,11)(9,30,33,45,32,42,39,18)(10,34,35,53,36,43,50,21)(17,46,56,31,44,59,48,58)(38,57,61,63,60,62,64,47);(1,3,15,42,7,19,25,45,6,16,29,30,8,27,5,18)(2,9,24,41,12,33,37,40,13,32,55,49,26,39,11,14)(4,17,43,62,23,48,53,63,22,44,34,57,28,56,21,47)(10,31,51,61,35,59,52,60,36,58,54,64,50,46,20,38);(3,16)(17,44)(18,45)(19,27)(30,42)(47,63)(48,56)(57,62);(1,4,15,43,7,23,25,53,6,22,29,34,8,28,5,21)(2,10,24,51,12,35,37,52,13,36,55,54,26,50,11,20)(3,17,42,62,19,48,45,63,16,44,30,57,27,56,18,47)(9,31,41,61,33,59,40,60,32,58,49,64,39,46,14,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=eh,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,h^a=hi,c^b=cg,f^b=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72225001, #72225009, …
maximal normal subgroup of	#72217287, #72225001, #72225009, …
maximal quotient of	C₂×#505704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412900);