#391431

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
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 ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

684 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298627);