#10451584

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

3953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 10358780);