#7641970

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×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6932 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)(8,15)(12,17)(14,19)(18,20)(21,22,27,31)(23,25,29,36)(24,30,26,28)(32,35,33,34);(1,2,3,4)(6,10)(7,13)(12,17)(14,19)(21,23)(22,25)(24,32)(26,33)(27,29)(28,34)(30,35)(31,36);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,24,27,26)(22,28,31,30)(23,32,29,33)(25,34,36,35);(21,24,27,26)(22,28,31,30)(23,32,29,33)(25,34,36,35);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,25)(22,29)(23,31)(24,34)(26,35)(27,36)(28,33)(30,32);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,26,27,24)(22,30,31,28)(23,33,29,32)(25,35,36,34)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=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, 7549166);