#382146

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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

788 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289342);