#8893591

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

4085 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)(3,11,13,8)(4,9,14,16)(5,12,15,10)(18,22)(23,25)(24,29)(30,31)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,7)(4,14)(8,11)(10,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(2,7)(5,15)(8,11)(9,16)(20,28)(24,29)(26,32)(30,31)(35,42)(39,46)(43,47)(44,48);(1,3,6,13)(2,8,7,11)(4,10,14,12)(5,9,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,4,6,14)(2,9,7,16)(3,12,13,10)(5,11,15,8)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46);(1,5)(2,10)(3,9)(4,11)(6,15)(7,12)(8,14)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800787);