#8425351

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	#38543	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

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, 8332547);