#7641920

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

6954 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)(23,29)(26,33)(30,35)(34,36);(1,2)(3,4)(6,10)(7,13)(12,17)(14,19)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35)

Polycyclic presentation:

2^9|b^2=h,c^2=i,e^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, 7549116);