#8434139

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

4263 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(20,28)(24,31)(27,32)(29,30)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341335);