#7795755

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4562 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=gi,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, 7702951);