#8730769

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	#37903	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8637965);