#242338

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(10,13)(11,14)(15,16)(17,18,21,25)(19,26,28,23)(20,24,22,30)(27,31,29,32);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,29)(21,28)(22,27)(24,31)(25,26)(30,32);(9,11,12,15)(10,14,13,16)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(17,20,21,22)(18,24,25,30)(19,27,28,29)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149534);