#245709

order	512
nilpotency class	3
isoclinism	isoclinic class of #245615
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1348 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312026, #72312041, #72312060
maximal normal subgroup of	#72312026, #72312041, #72312060

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152905);