#7813136

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
prime radical	#42999	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4568 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^2=gh,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, 7720332);