#111739

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#2541	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,4)(9,10,13,15,12,11,16,14)(17,18)(19,22)(20,23)(21,24);(2,4)(5,6)(7,8)(13,16)(14,15)(18,22)(19,20)(23,24);(6,8)(9,11)(10,12)(13,14)(15,16)(18,23)(19,20)(22,24)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217667
maximal normal subgroup of	#72217667
maximal quotient of	#72216328, #72216332, #72216337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18935);