#250253

order	512
nilpotency class	3
isoclinism	isoclinic class of #250199
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

1076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157449);