#7848835

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

4557 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)(9,10,15,17)(11,19,20,16)(12,18,21,24)(13,22,23,14)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(1,2,5,6)(3,7,4,8)(9,11,15,20)(10,16,17,19)(12,13,21,23)(14,24,22,18)(25,27)(26,31)(28,32)(29,37)(30,35)(33,40)(34,38)(36,39);(10,17)(11,20)(12,21)(14,22)(25,26,30,28)(27,32,35,31)(29,36,38,33)(34,39,37,40);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,13)(14,16)(15,21)(17,24)(19,22)(20,23)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(1,3,5,4)(2,7,6,8)(9,13)(10,14)(11,21)(12,20)(15,23)(16,24)(17,22)(18,19)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(1,4,5,3)(2,8,6,7)(9,14)(10,13)(11,18)(12,16)(15,22)(17,23)(19,21)(20,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7756031);