#250554

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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
prime radical	#9996	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1015 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157750);