#7623203

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

2444 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)(6,11)(8,19)(9,20)(12,21)(13,22)(14,26)(15,25)(17,32)(18,31)(23,24)(27,30)(28,33)(29,36)(34,35);(5,6,12,26,9,14,8,11,10,15,19,24,20,23,21,25)(7,13,27,36,18,29,17,22,16,28,32,35,31,34,30,33);(1,2,3,4);(5,7)(6,13)(8,17)(9,18)(10,16)(11,22)(12,27)(14,29)(15,28)(19,32)(20,31)(21,30)(23,34)(24,35)(25,33)(26,36);(7,16)(13,28)(17,30)(18,31)(22,33)(27,32)(29,34)(35,36)

Polycyclic presentation:

2^9|b^2=fhi,c^2=g,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530399);