#245835

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1365 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,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,d^b=dh,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257606, #72257626, #72257669
maximal normal subgroup of	#72257606, #72257626, #72257669
maximal quotient of	#72257946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153031);