#7798070

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	#48415	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7705266);