#7156216

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1928 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)(9,10,16,20,15,21,14,19)(11,12,24,17,25,28,23,27)(13,18,32,36,30,34,31,35)(22,26,38,33,37,39,29,40);(2,5)(3,7)(9,11)(10,17)(12,20)(13,29)(14,24)(15,25)(16,23)(18,26)(19,28)(21,27)(22,31)(30,38)(32,37)(33,35)(34,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,27,15,28,16,17)(10,11,20,24,21,25,19,23)(13,26,31,40,30,39,32,33)(18,22,36,38,34,37,35,29);(2,5)(6,8)(9,13,16,32,15,30,14,31)(10,18,20,36,21,34,19,35)(11,22,24,38,25,37,23,29)(12,26,17,33,28,39,27,40);(1,3,4,7)(2,6,5,8)(13,30)(18,34)(22,37)(26,39)(29,38)(31,32)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=h,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^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, 7063412);