#8097014

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	#45816	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4149 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004210);