#8755015

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8662211);