#347067

order	512
nilpotency class	3
isoclinism	isoclinic class of #347058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2497 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,e^c=ei,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234568
maximal normal subgroup of	#72234568

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254263);