#8492797

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=g,e^2=gi,f^2=g,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, 8399993);