#6782851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6690047);