#8096727

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4169 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=gh,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, 8003923);