#8095999

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,5,9,7)(4,8,12,15)(11,13,16,14)(17,18,24,26)(19,20,23,22)(21,29,32,25)(27,31,30,28)(34,40)(36,41)(42,48)(43,46);(4,12)(8,15)(11,16)(13,14)(17,19,24,23)(18,22,26,20)(21,30,32,27)(25,28,29,31)(33,34,39,40)(35,41,45,36)(37,42,44,48)(38,43,47,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,24,22)(18,19,26,23)(21,28,32,31)(25,27,29,30)(33,35,39,45)(34,41,40,36)(37,38,44,47)(42,43,48,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,32)(26,29)(33,36)(34,35)(37,46)(38,42)(39,41)(40,45)(43,44)(47,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,22,24,20)(18,19,26,23)(21,31,32,28)(25,27,29,30)(33,37)(34,42)(35,38)(36,46)(39,44)(40,48)(41,43)(45,47);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,15)(12,14)(17,23,24,19)(18,22,26,20)(21,30,32,27)(25,31,29,28)(33,38)(34,43)(35,44)(36,42)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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, 8003195);