#245353

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,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, 152549);