#7200504

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,38)(29,34)(30,35)(31,32)(33,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,31)(26,32,35,36)(28,37,29,39)(33,38,40,34);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,29)(26,33,35,40)(27,37,31,39)(32,38,36,34);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,29,30,28)(26,34,35,38)(27,37,31,39)(32,40,36,33);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(26,35)(28,29)(32,36)(37,39)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308866, #72309006, #72309019
maximal normal subgroup of	#72308866, #72309006, #72309019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107700);