#366240

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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

758 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)(12,13)(14,16)(17,18,22,27,23,28,21,26)(19,24,30,38,31,39,35,43)(20,25,34,42,33,41,36,44)(29,37,46,48,45,47,32,40);(2,6)(7,8)(10,13)(14,15)(17,19,22,30,23,31,21,35)(18,24,27,38,28,39,26,43)(20,32,34,29,33,46,36,45)(25,40,42,37,41,48,44,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,25)(19,29)(21,36)(22,34)(23,33)(24,37)(26,44)(27,42)(28,41)(30,46)(31,45)(32,35)(38,48)(39,47)(40,43);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(20,33)(25,41)(29,45)(32,46)(34,36)(37,47)(40,48)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=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, 273436);