#300168

order	512
nilpotency class	3
isoclinism	isoclinic class of #300141
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1224 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:

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

Polycyclic presentation:

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

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, 207364);