#7627232

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6486 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534428);