#7626418

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	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=h,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, 7533614);