#7815815

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

4559 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)(11,19)(12,13)(14,17)(15,18)(16,20)(21,24)(22,23)(25,26,30,29)(27,32,35,31)(28,36,38,33)(34,39,37,40);(1,2,4,5)(3,6,7,8)(9,11,15,20)(10,16,18,19)(12,21,22,17)(13,14,23,24)(25,27)(26,31)(28,34)(29,32)(30,35)(33,39)(36,40)(37,38);(11,20)(14,24)(16,19)(17,21)(25,27,30,35)(26,31,29,32)(28,34,38,37)(33,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,15,22)(10,13,18,23)(11,21,20,17)(14,19,24,16)(25,27)(26,32)(28,37)(29,31)(30,35)(33,39)(34,38)(36,40);(9,13)(10,12)(11,14)(15,23)(16,21)(17,19)(18,22)(20,24)(25,28,30,38)(26,33,29,36)(27,34,35,37)(31,39,32,40);(9,14)(10,17)(11,13)(12,19)(15,24)(16,22)(18,21)(20,23)(25,29,30,26)(27,31,35,32)(28,36,38,33)(34,39,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=h,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, 7723011);