#7776133

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	#59947	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4542 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=ghi,e^2=gi,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, 7683329);