#354904

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
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	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=eh,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262100);