#392664

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4250 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,b^a=be,c^a=cf,d^a=dg,f^a=fi,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299860);