#372240

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=f,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279436);