#8436212

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	#37496	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8343408);