#8506318

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	#43283	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,7)(4,5)(6,9)(8,12)(10,15)(11,13)(14,16)(17,18,23,28)(19,25,20,24)(21,29,31,26)(22,27,32,30)(36,44)(40,45)(43,48)(46,47);(1,3,6,12)(2,7,9,8)(4,13,14,10)(5,11,16,15)(17,19,23,20)(18,24,28,25)(21,30,31,27)(22,29,32,26)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,46)(47,48);(1,3)(2,8)(4,10)(5,11)(6,12)(7,9)(13,14)(15,16)(17,19,23,20)(18,25,28,24)(21,27,31,30)(22,29,32,26)(33,35)(34,37)(36,43)(38,41)(39,42)(40,46)(44,48)(45,47);(1,3)(2,7)(4,10)(5,15)(6,12)(8,9)(11,16)(13,14)(17,20,23,19)(18,25,28,24)(21,30,31,27)(22,29,32,26)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,4)(2,5)(3,10)(6,14)(7,15)(8,11)(9,16)(12,13)(17,21,23,31)(18,26,28,29)(19,27,20,30)(22,24,32,25)(33,36,38,44)(34,40,42,45)(35,43,41,48)(37,46,39,47);(1,5)(2,4)(3,11)(6,16)(7,13)(8,10)(9,14)(12,15)(17,22)(18,27)(19,29)(20,26)(21,24)(23,32)(25,31)(28,30)(33,37,38,39)(34,41,42,35)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413514);