#7842445

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	#51896	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4590 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)(9,10,15,13)(11,19,20,16)(12,17,21,22)(14,18,24,23)(27,29)(32,34)(35,36)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,20)(10,16,13,19)(12,14,21,24)(17,23,22,18)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(11,20)(14,24)(16,19)(18,23)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(9,12)(10,17)(11,14)(13,22)(15,21)(16,23)(18,19)(20,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(1,3,5,4)(2,7,6,8)(9,13,15,10)(11,19,20,16)(12,17,21,22)(14,23,24,18)(25,29,31,27)(26,32,30,34)(28,35,38,36)(33,40,37,39);(1,4,5,3)(2,8,6,7)(9,14)(10,18)(11,12)(13,23)(15,24)(16,22)(17,19)(20,21)(25,30,31,26)(27,32,29,34)(28,37,38,33)(35,39,36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=gh,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, 7749641);