#8097005

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	#45602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18)(19,22)(20,24)(21,25)(23,28)(26,30)(27,31)(29,32)(33,34,39,42)(35,36,37,40)(38,41,46,48)(43,44,45,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30)(33,35,39,37)(34,40,42,36)(38,43,46,45)(41,47,48,44);(17,20)(18,23)(19,26)(21,28)(22,29)(24,25)(27,32)(30,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,30)(25,31)(28,32)(33,36,39,40)(34,35,42,37)(38,44,46,47)(41,43,48,45);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,25)(27,32)(29,31)(33,37,39,35)(34,40,42,36)(38,43,46,45)(41,44,48,47);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32)(33,38,39,46)(34,41,42,48)(35,43,37,45)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004201);