#250098

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
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	#18870	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157294);