#7920002

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	#36438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4367 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)(9,10,15,13)(11,16,22,19)(12,23,21,17)(14,18,20,24)(25,26)(27,34)(28,37)(29,31)(30,33)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,19)(14,21)(15,22)(17,18)(23,24)(25,27)(26,32)(28,38)(29,34)(30,35)(31,36)(33,40)(37,39);(1,3,4,7)(2,6,5,8)(9,10,15,13)(11,19,22,16)(12,17,21,23)(14,18,20,24)(25,28,31,30)(26,33,29,37)(27,35,36,38)(32,39,34,40);(9,12,15,21)(10,17,13,23)(11,20,22,14)(16,18,19,24)(25,29,31,26)(27,34,36,32)(28,33,30,37)(35,39,38,40);(9,13,15,10)(11,19,22,16)(12,23,21,17)(14,18,20,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,36)(33,39)(37,40);(9,14,15,20)(10,18,13,24)(11,21,22,12)(16,17,19,23)(25,30,31,28)(26,33,29,37)(27,35,36,38)(32,40,34,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=i,e^2=hi,f^2=i,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, 7827198);