#7737036

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	#48741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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)(27,29)(28,35)(30,32)(31,33)(34,37)(36,38)(39,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27)(26,28)(29,33)(30,38)(31,35)(32,37)(34,39)(36,40);(10,14)(11,17)(16,19)(18,23)(25,28,31,29)(26,27,33,35)(30,36,39,37)(32,34,40,38);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,31,28)(26,27,33,35)(30,37,39,36)(32,34,40,38);(1,2,5,4)(3,6,8,7)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30,31,39)(26,32,33,40)(27,34,35,38)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644232);