#356615

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

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 ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1984 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)(12,20,21,18)(13,19)(15,23)(22,24);(2,4)(5,6,8,7);(5,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263811);