#8093421

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

4106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000617);