#417528

order	512
nilpotency class	3
isoclinism	isoclinic class of #417496
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225689, #72251366, #72251367, …
maximal normal subgroup of	#72225689, #72251366, #72251367, …
maximal quotient of	#72225659, #72233896, #72251507

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324724);