#7813808

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	#51084	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=gh,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, 7721004);