#7806298

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6760 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=h,e^2=i,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, 7713494);