#7626257

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#534	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6730 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)(7,16)(8,19)(9,10)(11,12)(13,17)(14,18)(15,20)(22,26)(27,32)(28,33)(34,36);(6,12)(7,17)(9,15)(14,19)(24,31)(28,33)(29,35)(34,36);(1,3,4,2)(5,7)(6,13)(8,15)(9,18)(10,19)(11,17)(12,16)(14,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,8,11,18)(6,14,12,19)(7,15,17,9)(10,13,20,16)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,9,11,15)(6,10,12,20)(7,18,17,8)(13,19,16,14)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36);(5,10)(6,15)(7,19)(8,13)(9,12)(11,20)(14,17)(16,18)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh

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, 7533453);