#356617

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1672 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)(9,10,13,12)(11,17,18,15)(14,20,21,16)(19,22,24,23);(9,11,14,19,13,18,21,24)(10,15,16,22,12,17,20,23);(2,4)(5,6,8,7)(9,12,13,10)(11,17,18,15)(14,20,21,16)(19,23,24,22);(5,7,8,6)(11,18)(14,21)(15,17)(16,20)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,h^2=i,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, 263813);