#7626448

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×#3511	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6632 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)(7,16)(8,9)(10,20)(11,13)(12,17)(14,19)(15,18)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7,11,17)(6,12,13,16)(8,18,19,10)(9,20,14,15)(21,22,25,28)(23,26,31,29)(24,27,32,34)(30,33,36,35);(1,3,4,2)(6,13)(7,17)(8,19)(15,20)(21,22)(23,26)(24,27)(25,28)(29,31)(30,33)(32,34)(35,36);(5,8)(6,14)(7,18)(9,13)(10,17)(11,19)(12,15)(16,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35);(5,9,11,14)(6,8,13,19)(7,15,17,20)(10,16,18,12)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35);(5,10,11,18)(6,15,13,20)(7,8,17,19)(9,16,14,12)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=g,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, 7533644);