#7773914

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	#39375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26)(27,29)(28,33)(30,32)(31,34)(35,38)(36,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,27,30,34)(26,31,32,29)(28,35,36,40)(33,37,39,38);(1,3,5,4)(2,7,6,8)(11,17)(15,19)(18,23)(22,24)(26,32)(27,34)(28,36)(37,38);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,29,30,31)(26,34,32,27)(28,37,36,38)(33,35,39,40);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=gh,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7681110);