#7626547

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×#3552	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6452 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)(7,10)(8,15)(9,11)(12,16)(13,19)(14,17)(18,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7,9,14)(6,10,11,17)(8,13,16,20)(12,18,15,19)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(6,11)(10,17)(12,15)(18,19)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,6)(7,10)(8,12)(9,11)(13,18)(14,17)(15,16)(19,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(1,3,4,2)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,19)(16,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35);(5,8,9,16)(6,12,11,15)(7,13,14,20)(10,18,17,19)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=gi,f^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, 7533743);