#250403

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
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

1025 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,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, 157599);