#7846938

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	#47849	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4409 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=g,e^2=gi,f^2=gi,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, 7754134);