#8097272

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	#45649	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)(3,8,13,12)(4,14,5,9)(10,16,11,15)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31)(33,34,39,44)(35,38,43,40)(36,41,46,48)(37,47,45,42);(1,3,7,13)(2,8,6,12)(4,10,5,11)(9,15,14,16)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28)(33,35)(34,40)(36,45)(37,46)(38,44)(39,43)(41,47)(42,48);(1,4,7,5)(2,9,6,14)(3,10,13,11)(8,15,12,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,36,39,46)(34,41,44,48)(35,45,43,37)(38,42,40,47);(1,3)(2,8)(4,10)(5,11)(6,12)(7,13)(9,15)(14,16)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29)(33,37,39,45)(34,42,44,47)(35,46,43,36)(38,41,40,48);(1,5,7,4)(2,9,6,14)(3,11,13,10)(8,15,12,16)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31)(33,35)(34,38)(36,45)(37,46)(39,43)(40,44)(41,42)(47,48);(1,6,7,2)(3,12,13,8)(4,14,5,9)(10,16,11,15)(17,22,23,30)(18,25,26,32)(19,27,21,29)(20,28,24,31)(33,38)(34,43)(35,44)(36,42)(37,41)(39,40)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,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, 8004468);