#250124

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
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	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

994 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157320);