#7813815

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	#60021	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,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, 7721011);