#7642957

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6898 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:

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

Polycyclic presentation:

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

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, 7550153);