#8492936

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400132);