#7917622

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	#45448	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4473 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824818);