#8505095

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=ghi,e^2=g,f^2=gi,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, 8412291);