#354275

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2216 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)(5,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(9,11)(10,14,16,17)(12,18)(13,19)(15,21,22,23)(20,24);(2,4)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23);(5,7,8,6)(12,20)(15,22)(18,24)(21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261471);