#7798051

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	#50746	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4552 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)(9,10,14,13)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,26)(27,35)(28,34)(29,31)(30,32)(33,38)(36,39)(37,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27,31,30)(26,32,29,35)(28,37,38,36)(33,40,34,39);(11,18)(15,17)(19,24)(22,23)(25,28,31,38)(26,33,29,34)(27,36,30,37)(32,39,35,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28)(26,34)(27,37)(29,33)(30,36)(31,38)(32,39)(35,40);(9,13,14,10)(11,15,18,17)(12,16,20,21)(19,23,24,22)(25,29,31,26)(27,32,30,35)(28,34,38,33)(36,39,37,40);(9,11)(10,17)(12,19)(13,15)(14,18)(16,23)(20,24)(21,22)(25,30,31,27)(26,32,29,35)(28,36,38,37)(33,40,34,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gi,e^2=i,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, 7705247);