#250693

order	512
nilpotency class	3
isoclinism	isoclinic class of #250647
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

1080 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=i,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,d^b=dh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157889);