#7627423

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3531	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6364 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534619);