#8425452

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4175 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332648);