#356494

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2122 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)(9,10,14,18,15,17,12,13)(11,19,20,21,22,16,24,23);(5,6,7,8)(9,11)(10,16)(12,20)(13,23)(14,24)(15,22)(17,19)(18,21);(2,4)(9,12,15,14)(10,13,17,18)(11,20,22,24)(16,23,19,21);(9,13,12,17,15,18,14,10)(11,21,24,19,22,23,20,16)

Polycyclic presentation:

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

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, 263690);