#8926708

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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)(3,7)(4,14)(5,9)(6,10)(8,12)(11,13)(15,16)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(34,39)(37,45)(40,46)(44,48);(3,11)(5,15)(7,13)(9,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(36,43)(40,46)(42,47)(44,48);(1,3,6,11)(2,7,10,13)(4,5,12,15)(8,16,14,9)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,4)(2,8)(3,5)(6,12)(7,16)(9,13)(10,14)(11,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,16)(13,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,36,38,43)(34,40,39,46)(35,42,41,47)(37,44,45,48);(1,2)(3,13)(4,8)(5,9)(6,10)(7,11)(12,14)(15,16)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833904);