#247949

order	512
nilpotency class	3
isoclinism	isoclinic class of #247879
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
prime radical	#24222	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1196 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)(3,5)(4,8)(9,10)(11,16)(12,21)(13,17)(14,22)(15,18)(19,23)(20,24);(1,2,6,4)(3,7,5,8)(9,11,15,20)(10,16,18,24)(12,13,19,14)(17,23,22,21);(1,3,6,5)(2,7,4,8)(9,12)(10,16)(11,13)(14,20)(15,19)(17,23)(18,24)(21,22);(1,4,6,2)(3,8,5,7)(9,13,15,14)(10,17,18,22)(11,19,20,12)(16,23,24,21)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155145);