#8926667

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

4115 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=gh,e^2=g,f^2=h,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, 8833863);