#7650704

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	#38077	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4565 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7557900);