#184585

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 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)(3,7)(4,6)(5,8)(17,18,20,24,21,23,28,31)(19,25,27,32,26,22,29,30);(1,3)(2,5)(4,8)(6,7)(9,10,11,13,12,14,15,16)(19,26)(22,25)(27,29)(30,32);(3,8)(5,7)(17,19,20,27,21,26,28,29)(18,22,24,30,23,25,31,32);(2,6)(5,7)(18,23)(19,26)(24,31)(27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=deh,c^b=ceh,d^c=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91781);