#265304

order	512
nilpotency class	3
isoclinism	isoclinic class of #265303
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=eh,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172500);