#7658953

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566149);