#8093233

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	#45510	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

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

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, 8000429);