#299243

order	512
nilpotency class	3
isoclinism	isoclinic class of #299233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206439);