#405698

order	512
nilpotency class	3
isoclinism	isoclinic class of #394273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1382 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,15)(14,16)(17,24)(18,22)(19,20)(21,23);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,19,21,17)(14,18)(15,20,23,24)(16,22);(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(1,4,5,3)(2,6,7,8)(9,13)(10,11)(12,20)(14,22)(15,17)(16,18)(19,23)(21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312894);