#393782

order	512
nilpotency class	3
isoclinism	isoclinic class of #195954
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300978);