#8927262

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

4320 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8834458);