#6792996

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285261
maximal normal subgroup of	#72285261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6700192);