#345356

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=e,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252552);