#8097205

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	#52970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=h,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, 8004401);