#7977505

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	#40524	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4560 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7884701);