#411731

order	512
nilpotency class	3
isoclinism	isoclinic class of #394061
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35000	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2324 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=h,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei

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, 318927);