#379348

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

774 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,24,30,22,28,23,29)(19,31,34,27,33,25,21,39)(20,26,38,45,37,44,40,46)(32,47,35,43,48,41,36,42);(2,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19,23,21,22,33,24,34)(18,25,29,27,28,31,30,39)(20,35,38,48,37,36,40,32)(26,42,45,47,44,43,46,41);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,15,16,14)(17,20,22,37)(18,26,28,44)(19,32,33,48)(21,36,34,35)(23,40,24,38)(25,41,31,47)(27,43,39,42)(29,46,30,45);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(17,21,23,33,22,34,24,19)(18,27,29,31,28,39,30,25)(20,36,40,48,37,35,38,32)(26,43,46,47,44,42,45,41)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,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, 286544);