#8436640

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	#40692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gh,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, 8343836);