#7626331

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×#3566	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6612 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)(5,6,11,8)(7,10,15,12)(9,13,17,16)(14,18,20,19)(21,22,27,25)(23,30,29,26)(24,32,34,28)(31,33,35,36);(9,17)(13,16)(14,20)(18,19)(22,25)(23,29)(28,32)(31,35);(5,7,11,15)(6,12,8,10)(9,14,17,20)(13,19,16,18)(21,23)(22,26)(24,31)(25,30)(27,29)(28,33)(32,36)(34,35);(1,2,4,3)(5,8,11,6)(7,10,15,12)(9,16,17,13)(14,18,20,19)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36);(1,3,4,2)(5,9,11,17)(6,13,8,16)(7,14,15,20)(10,18,12,19)(21,25,27,22)(23,30,29,26)(24,32,34,28)(31,36,35,33);(5,10)(6,7)(8,15)(9,18)(11,12)(13,14)(16,20)(17,19)(21,26,27,30)(22,29,25,23)(24,33,34,36)(28,35,32,31)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=gi,e^2=i,f^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, 7533527);