#8810188

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	#44098	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(2,6)(7,9)(8,13)(14,15)(21,29)(25,28)(27,30)(31,32)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(3,10)(4,12)(7,9)(8,13)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717384);