#8925040

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

4165 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)(3,9)(4,10)(5,15)(6,8)(7,13)(11,12)(14,16)(19,25)(23,30)(26,28)(31,32)(33,34,39,43)(35,40,45,42)(36,46,44,41)(37,38,47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,9)(10,16)(14,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,40)(36,47)(37,44)(38,41)(39,45)(42,43)(46,48);(2,8)(5,10)(7,9)(12,16)(18,22)(23,30)(24,27)(31,32)(33,36,39,44)(34,41,43,46)(35,37,45,47)(38,40,48,42);(1,3)(2,9)(4,14)(5,12)(6,13)(7,8)(10,16)(11,15)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,35)(34,42)(36,37)(38,41)(39,45)(40,43)(44,47)(46,48);(1,4)(2,10)(3,11)(5,8)(6,15)(7,16)(9,12)(13,14)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37)(34,38)(35,44)(36,45)(39,47)(40,41)(42,46)(43,48);(1,5,6,10)(2,4,8,15)(3,12,13,16)(7,11,9,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,38)(34,37)(35,41)(36,42)(39,48)(40,44)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832236);