#7798086

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	#49206	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 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)(9,10,14,16)(11,19,20,15)(12,17,18,13)(21,23,24,22)(25,26,31,28)(27,32,34,30)(29,33,39,36)(35,40,37,38);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,23)(14,20)(16,19)(17,22)(18,24)(25,27,31,34)(26,32,28,30)(29,37,39,35)(33,38,36,40);(1,3,5,4)(2,6,8,7)(10,16)(11,20)(12,18)(22,23)(27,34)(30,32)(35,37)(38,40);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,21)(13,16)(14,18)(15,22)(19,23)(20,24)(25,28,31,26)(27,30,34,32)(29,33,39,36)(35,40,37,38);(9,13,14,17)(10,18,16,12)(11,22,20,23)(15,21,19,24)(25,29,31,39)(26,33,28,36)(27,35,34,37)(30,38,32,40);(1,4,5,3)(2,7,8,6)(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,23)(18,24)(25,30)(26,27)(28,34)(29,38)(31,32)(33,35)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gi,e^2=i,f^2=gh,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, 7705282);