#7627626

order	512
nilpotency class	2
isoclinism	isoclinic class of #62972
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×#3536	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6750 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534822);