#7795730

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	#44328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4377 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)(9,10,15,12)(11,19,20,16)(13,17,24,21)(14,18,23,22)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,6)(4,7)(5,8)(9,11,15,20)(10,16,12,19)(13,23,24,14)(17,18,21,22)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(11,20)(14,23)(16,19)(18,22)(25,26,30,34)(27,31,28,32)(29,33,37,40)(35,38,36,39);(1,3,5,4)(2,6,8,7)(9,12,15,10)(11,19,20,16)(13,17,24,21)(14,22,23,18)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,14)(12,21)(15,24)(16,22)(18,19)(20,23)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(9,14)(10,18)(11,13)(12,22)(15,23)(16,21)(17,19)(20,24)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=gh,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, 7702926);