#7624897

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 7532093);