#8187001

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
prime radical	#38488	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(34,39)(35,41)(36,43)(44,48);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(6,14)(9,12)(11,13)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35)(34,37)(36,42)(38,41)(39,45)(40,44)(43,47)(46,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094197);