#7805058

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	#41404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7712254);