#184479

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

700 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)(9,10,11,14,12,13,15,16);(2,5)(3,7)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,29)(31,32);(2,5)(6,8)(19,26)(22,25)(27,29)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=de,c^b=cei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91675);