#7770307

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4505 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)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26)(27,32)(28,37)(29,34)(30,35)(31,38)(33,39)(36,40);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(11,17)(15,18)(19,23)(22,24)(25,27,30,38)(26,31,35,32)(28,36,39,34)(29,33,40,37);(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,27)(26,32)(28,34)(29,37)(30,38)(31,35)(33,40)(36,39);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,28)(26,33)(27,36)(29,32)(30,39)(31,40)(34,38)(35,37);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,29)(26,34)(27,37)(28,32)(30,40)(31,39)(33,38)(35,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=hi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7677503);