#494012

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1250 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,15)(14,23,22,18)(16,20,24,21);(2,5)(6,8)(10,15)(11,12)(18,23)(20,21);(9,11,13,12)(10,16,15,24)(14,20,22,21)(17,18,19,23);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,17,15,19)(14,21,22,20)(16,18,24,23)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=e,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=di,g^c=gh,g^f=gi

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, 401208);