#7622826

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

3992 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^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.

Relationships

maximal subgroup of	#72265540
maximal normal subgroup of	#72265540
maximal quotient of	#72310184, D₄×#2187, #72310196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530022);