#8927315

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

4198 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,34)(35,38)(36,45)(37,41)(39,42)(40,44)(43,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,18,21,24)(19,22,25,26)(20,23,28,30)(27,29,31,32)(33,35)(34,38)(36,37)(39,43)(40,46)(41,45)(42,48)(44,47);(2,7)(4,13)(6,8)(10,16)(19,25)(20,28)(22,26)(23,30)(35,43)(36,44)(38,48)(40,45);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,19,21,25)(18,22,24,26)(20,27,28,31)(23,29,30,32)(33,36)(34,40)(35,37)(38,46)(39,44)(41,48)(42,45)(43,47);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,37)(34,41)(35,44)(36,43)(38,40)(39,47)(42,46)(45,48);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,20,21,28)(18,23,24,30)(19,27,25,31)(22,29,26,32)(33,38)(34,35)(36,46)(37,40)(39,48)(41,44)(42,43)(45,47)

Polycyclic presentation:

2^9|b^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, 8834511);