#7712230

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

6328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7619426);