#356183

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

1976 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:

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

Polycyclic presentation:

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

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, 263379);