#7622942

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530138);