#269148

order	512
nilpotency class	3
isoclinism	isoclinic class of #269147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

921 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^b=fh,g^b=gh,e^c=eh,f^c=fi,g^c=gh,f^d=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, 176344);