#323723

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=egi,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230919);