#7053859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fg,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313310, #72313612, #72313904
maximal normal subgroup of	#72313310, #72313612, #72313904

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961055);