#250468

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1011 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157664);