#7623205

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
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

2456 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)(8,12)(9,19)(13,21)(14,22)(15,29)(17,25)(18,31)(20,26)(23,24)(27,32)(28,36)(30,35)(33,34);(5,6,12,24,9,15,20,21,10,13,26,29,19,23,8,11)(7,14,25,34,18,28,30,32,16,27,35,36,31,33,17,22);(1,2,3,4)(6,13)(11,21)(14,27)(15,23)(22,32)(24,29)(28,33)(34,36);(5,7)(6,14)(8,17)(9,18)(10,16)(11,22)(12,25)(13,27)(15,28)(19,31)(20,30)(21,32)(23,33)(24,34)(26,35)(29,36);(7,16)(14,27)(17,30)(18,31)(22,32)(25,35)(28,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322758
maximal normal subgroup of	#72322758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530401);