#9794334

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

3993 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9701530);