#8464184

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	#42945	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=gh,e^2=i,f^2=gi,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, 8371380);