#7624647

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1353 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318797, #72320309, #72320623
maximal normal subgroup of	#72318797, #72320309, #72320623

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531843);