#7626521

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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

6472 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)(6,10)(8,16)(11,13)(14,20)(21,22,27,24)(23,30,25,28)(26,32,34,29)(31,35,33,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22,27,24)(23,28,25,30)(26,29,34,32)(31,35,33,36);(5,6,9,10)(7,11,15,13)(8,12,16,17)(14,18,20,19)(21,23,27,25)(22,28,24,30)(26,31,34,33)(29,35,32,36);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,27,22)(23,28,25,30)(26,32,34,29)(31,35,33,36);(1,3,4,2)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25,27,23)(22,28,24,30)(26,33,34,31)(29,35,32,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26)(22,29)(23,31)(24,32)(25,33)(27,34)(28,35)(30,36)

Polycyclic presentation:

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

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, 7533717);