#7623983

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1389 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343987
maximal normal subgroup of	#72343987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531179);