#7200523

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

1506 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)(25,26,30,35)(27,37,38,32)(28,39,40,34)(29,36,31,33);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,38)(26,32,35,37)(28,29,40,31)(33,34,36,39);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,38)(29,31)(32,39)(34,37)(35,36);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,29)(30,40)(31,38)(32,36)(33,37)(35,39);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,33,35,36)(27,28,38,40)(32,39,37,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^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	#72308774, #72308971, #72309019
maximal normal subgroup of	#72308774, #72308971, #72309019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107719);