#8533317

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	#56325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8440513);