#245518

order	512
nilpotency class	3
isoclinism	isoclinic class of #245447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

969 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 969 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,11)(3,8,12,9)(4,13,14,16)(5,10,6,15)(17,18,22,27)(19,28,32,29)(20,34,36,42)(21,37,23,26)(24,33,40,31)(25,38,41,35)(30,39,46,48)(43,47,44,45);(1,3,7,12)(2,8,11,9)(4,5,14,6)(10,16,15,13)(17,19,22,32)(18,24,27,40)(20,30,36,46)(21,33,23,31)(25,39,41,48)(26,29,37,28)(34,43,42,44)(35,47,38,45);(1,4,7,14)(2,9,11,8)(3,5,12,6)(10,16,15,13)(17,18)(19,29)(20,25)(21,26)(22,27)(23,37)(24,31)(28,32)(30,44)(33,40)(34,35)(36,41)(38,42)(39,47)(43,46)(45,48);(1,5,7,6)(2,10,11,15)(3,4,12,14)(8,13,9,16)(17,20,22,36)(18,25,27,41)(19,30,32,46)(21,38,23,35)(24,39,40,48)(26,42,37,34)(28,43,29,44)(31,47,33,45)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152714);