#184382

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,g^2=i,b^a=be,c^a=ci,c^b=ch

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91578);