#245723

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1530 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)(3,5)(4,7)(6,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,20)(21,24)(22,23);(2,5)(3,8)(6,7)(9,11,14,21)(10,15,18,23)(12,22,13,20)(16,24,17,19);(10,16)(12,13)(15,19)(17,18)(20,22)(23,24);(2,5)(3,8)(6,7)(10,17)(12,13)(15,19)(16,18)(20,22)(23,24)

Polycyclic presentation:

2^9|b^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280008
maximal normal subgroup of	#72280008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152919);