#7627431

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×#3539	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6774 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=gi,f^2=g,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, 7534627);