#7626414

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×#3513	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6834 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)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(21,22,27,28)(23,24,25,29)(26,34,35,30)(31,36,33,32);(7,13)(11,17)(14,19)(18,20)(22,28)(23,25)(30,34)(31,33);(1,3,4,2)(6,10)(11,17)(12,15)(18,20)(21,23,27,25)(22,29,28,24)(26,31,35,33)(30,36,34,32);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,24,27,29)(22,23,28,25)(26,32,35,36)(30,31,34,33);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,25,27,23)(22,29,28,24)(26,33,35,31)(30,36,34,32);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,26,27,35)(22,30,28,34)(23,31,25,33)(24,32,29,36)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,e^2=h,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, 7533610);