#7622829

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3875 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:

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

Polycyclic presentation:

2^9|c^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,f^b=fh,h^b=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, 7530025);