#7813799

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

4484 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)(25,26,30,29)(27,32,35,31)(28,36,38,33)(34,39,37,40);(1,2,5,6)(3,7,4,8)(9,11,14,20)(10,15,18,19)(12,21,22,13)(16,17,23,24)(25,27)(26,31)(28,34)(29,32)(30,35)(33,39)(36,40)(37,38);(11,20)(13,21)(15,19)(17,24)(25,27,30,35)(26,31,29,32)(28,34,38,37)(33,39,36,40);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,18,23)(11,21,20,13)(15,17,19,24)(25,27)(26,32)(28,37)(29,31)(30,35)(33,39)(34,38)(36,40);(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,28,30,38)(26,33,29,36)(27,34,35,37)(31,39,32,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,22,20,12)(15,16,19,23)(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=g,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, 7720995);