#7643229

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×#3493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6782 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)(9,17)(14,18)(15,20)(16,19)(22,23)(25,33)(27,29)(31,36);(1,2)(3,4)(5,6)(7,12)(8,13)(9,14)(10,11)(15,19)(16,20)(17,18)(21,22,26,23)(24,27,32,29)(25,28,33,30)(31,34,36,35);(6,11)(8,12)(14,18)(16,19)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34);(5,7,10,13)(6,12,11,8)(9,15,17,20)(14,19,18,16)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34);(5,8,10,12)(6,13,11,7)(9,16,17,19)(14,20,18,15)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,9)(6,14)(7,15)(8,16)(10,17)(11,18)(12,19)(13,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,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, 7550425);