#245372

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
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	#61807	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

950 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)(3,10)(4,6)(5,9)(7,11)(8,14)(12,16)(13,15)(17,18,22,26)(19,23,29,38)(20,30,31,41)(21,33,35,25)(24,34,40,32)(27,42,43,48)(28,45,47,37)(36,46,39,44);(2,6)(4,14)(7,10)(8,9)(11,13)(15,16)(17,19,22,29)(18,23,26,38)(20,27,31,43)(21,28,35,47)(24,39,40,36)(25,37,33,45)(30,48,41,42)(32,44,34,46);(1,2,5,9)(3,7,12,15)(4,8,14,6)(10,13,16,11)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,41)(26,40)(28,46)(29,43)(30,33)(32,35)(37,48)(38,39)(42,45)(44,47);(1,3,5,12)(2,7,9,15)(4,13,14,11)(6,10,8,16)(21,35)(25,33)(28,47)(30,41)(32,34)(37,45)(42,48)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=h,f^2=i,g^2=h,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=gh,e^c=ei,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152568);