#7791003

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

4558 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=hi,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, 7698199);