#7156232

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61081	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8)(9,10,16,22,15,20,14,21)(11,23,28,17,26,38,27,35)(12,29,13,19,30,37,31,18)(24,36,32,39,25,33,40,34);(2,5)(3,7)(9,11)(10,17)(12,25)(13,32)(14,28)(15,26)(16,27)(18,36)(19,33)(20,35)(21,38)(22,23)(24,30)(29,34)(31,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12,16,13,15,30,14,31)(10,18,21,37,20,19,22,29)(11,24,27,40,26,25,28,32)(17,33,38,34,35,36,23,39);(2,5)(6,8)(9,13,16,30,15,31,14,12)(10,19,22,37,20,18,21,29)(11,25,28,40,26,24,27,32)(17,34,38,36,35,39,23,33);(1,3,4,7)(2,6,5,8)(10,20)(11,26)(12,30)(13,31)(21,22)(27,28)(33,36)(34,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7063428);