#379376

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,28,22,27,29,38)(19,24,32,41,31,40,21,26)(20,33,37,43,35,25,36,44)(30,45,47,42,46,39,34,48);(1,3,5,4)(2,6,8,7)(10,13)(14,15)(17,19,29,21,22,31,23,32)(18,24,38,26,27,40,28,41)(20,34,37,30,35,47,36,46)(25,42,44,39,33,48,43,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,39)(26,42)(27,33)(28,44)(29,36)(31,46)(32,47)(38,43)(40,45)(41,48);(1,4,5,3)(2,7,8,6)(10,13)(14,15)(17,21,29,31,22,32,23,19)(18,26,38,40,27,41,28,24)(20,34,36,46,35,47,37,30)(25,42,43,45,33,48,44,39)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=e,d^2=efh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286572);