#6352129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36029	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3007 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259325);