#8097208

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

4017 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004404);