#7712458

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	#38034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4489 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)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26,31,28)(27,35,36,32)(29,33,39,38)(30,37,40,34);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,36)(26,32,28,35)(29,37,39,34)(30,38,40,33);(10,15)(11,18)(16,20)(19,24)(27,36)(29,39)(32,35)(33,38);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)(25,28,31,26)(27,35,36,32)(29,38,39,33)(30,37,40,34);(1,4,5,3)(2,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,29,31,39)(26,33,28,38)(27,37,36,34)(30,35,40,32);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,30,31,40)(26,34,28,37)(27,38,36,33)(29,35,39,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7619654);