#8455706

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=h,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, 8362902);