#7799541

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	#59755	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 7706737);