#8463647

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	#53930	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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370843);