#7715994

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	#48813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4339 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7623190);