#7644134

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
prime radical	C₂×#3506	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6772 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)(5,6,10,8)(7,13,14,11)(9,16,17,12)(15,18,20,19)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(1,2,3,4)(7,14)(11,13)(15,20)(18,19)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(6,8)(11,13)(12,16)(18,19)(23,24)(27,30)(31,32)(35,36);(5,7,10,14)(6,11,8,13)(9,15,17,20)(12,18,16,19)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(5,8,10,6)(7,13,14,11)(9,16,17,12)(15,19,20,18)(21,24,26,23)(22,27,29,30)(25,32,34,31)(28,35,33,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551330);