#8493229

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

4247 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400425);