#8461708

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	#57437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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)(4,9)(5,10)(6,11)(8,13)(12,14)(15,16)(17,18,24,27)(19,28,31,25)(20,23,21,32)(22,30,29,26)(36,44)(41,45)(42,48)(46,47);(1,3,6,13)(2,7,11,8)(4,10,14,16)(5,12,15,9)(17,19,24,31)(18,25,27,28)(20,29,21,22)(23,30,32,26)(33,34,38,37)(35,40,43,39)(36,41,44,45)(42,47,48,46);(1,3)(2,8)(4,10)(5,12)(6,13)(7,11)(9,15)(14,16)(17,20,24,21)(18,23,27,32)(19,29,31,22)(25,30,28,26)(33,35)(34,39)(36,42)(37,40)(38,43)(41,46)(44,48)(45,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,15)(14,16)(17,21,24,20)(18,23,27,32)(19,22,31,29)(25,30,28,26)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,48)(45,46);(1,4,6,14)(2,9,11,12)(3,10,13,16)(5,8,15,7)(17,22)(18,26)(19,20)(21,31)(23,25)(24,29)(27,30)(28,32)(33,36,38,44)(34,41,37,45)(35,42,43,48)(39,46,40,47);(1,5)(2,10)(3,12)(4,8)(6,15)(7,14)(9,13)(11,16)(17,23)(18,21)(19,30)(20,27)(22,25)(24,32)(26,31)(28,29)(33,37,38,34)(35,40,43,39)(36,45,44,41)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=i,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, 8368904);