#10368646

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275842);