#265328

order	512
nilpotency class	3
isoclinism	isoclinic class of #265327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1125 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)(4,6)(7,8)(9,10,15,11)(12,20,23,24)(13,14,19,16)(17,21,18,22);(1,3)(2,5)(4,7)(6,8)(9,11,14,19,15,10,16,13)(12,18,21,20,23,17,22,24);(5,8)(9,12)(10,17)(11,18)(13,24)(14,21)(15,23)(16,22)(19,20);(9,13,14,11,15,19,16,10)(12,20,21,17,23,24,22,18)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=e,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,d^b=dh,f^b=fi,d^c=dh,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172524);