#7623852

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
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

1308 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216687, #72225003, #72225015, …
maximal normal subgroup of	#72216687, #72225003, #72225015, …
maximal quotient of	C₂×#7623852, #72319560, #72321861

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531048);