#7518959

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(9,10)(11,17)(12,21)(13,15)(14,20)(16,18)(19,23)(22,24);(1,2)(3,4)(5,6)(7,8)(9,11,18,24,13,19,12,20)(10,14,21,23,15,22,16,17);(6,8)(10,15)(11,19)(16,21)(20,24);(2,4)(6,8)(10,15)(14,22)(16,21)(17,23);(9,11,18,24,13,19,12,20)(10,14,21,23,15,22,16,17)

Polycyclic presentation:

2^9|b^2=fi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283896, #72284414
maximal normal subgroup of	#72283896, #72284414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426155);