#250774

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
prime radical	#18222	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1016 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,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,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, 157970);