#8463622

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	#36839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30)(33,34,38,42)(35,43,45,39)(36,40,46,48)(37,41,47,44);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(18,23)(19,26)(24,30)(27,32)(33,35,38,45)(34,39,42,43)(36,44,46,41)(37,40,47,48);(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,47)(46,48);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31)(33,34)(35,43)(36,40)(37,44)(38,42)(39,45)(41,47)(46,48);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36,38,46)(34,40,42,48)(35,44,45,41)(37,39,47,43);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)(33,37)(34,41)(35,40)(36,39)(38,47)(42,44)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,e^2=hi,f^2=ghi,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, 8370818);