#303297

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#39420	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1089 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,26)(20,24)(21,25)(22,29)(23,27)(28,32)(30,31);(1,3,5,4)(2,6,8,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,26)(20,23)(21,25)(22,30)(24,27)(28,32)(29,31);(1,3,5,4)(2,7,8,6)(9,11,12,10,14,16,13,15)(17,19,27,30,21,28,20,29)(18,22,24,32,25,31,23,26);(1,4,5,3)(2,7,8,6)(9,12,14,13)(10,11,15,16)(19,28)(22,31)(26,32)(29,30)

Polycyclic presentation:

2^9|b^2=e,c^2=efh,d^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210493);