#354233

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2012 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)(10,15)(12,13)(16,23)(17,20)(18,21)(22,24);(5,6,7,8)(9,10)(11,16)(12,15)(13,20)(14,17)(18,22)(19,24)(21,23);(9,11,14,19)(10,16,17,24)(12,18,13,21)(15,22,20,23);(1,3,4,2)(9,12,14,13)(10,15,17,20)(11,18,19,21)(16,22,24,23)

Polycyclic presentation:

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

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, 261429);