#7897738

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	#45833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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)(9,10)(11,19)(12,22)(13,17)(14,18)(15,21)(16,23)(20,24)(25,26,31,35)(27,34,29,32)(28,37,30,33)(36,39,38,40);(1,2,5,6)(3,7,4,8)(9,11,14,21)(10,15,18,19)(12,20,23,13)(16,24,22,17)(25,27,31,29)(26,32,35,34)(28,38,30,36)(33,40,37,39);(9,10,14,18)(11,15,21,19)(12,16,23,22)(13,17,20,24)(25,28,31,30)(26,33,35,37)(27,36,29,38)(32,39,34,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,24)(22,23)(25,29,31,27)(26,34,35,32)(28,36,30,38)(33,39,37,40);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,20)(13,21)(14,23)(15,24)(17,19)(18,22)(25,26)(27,32)(28,33)(29,34)(30,37)(31,35)(36,39)(38,40);(9,13,14,20)(10,17,18,24)(11,12,21,23)(15,16,19,22)(25,30,31,28)(26,33,35,37)(27,36,29,38)(32,40,34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7804934);