#8731135

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8638331);