#7627664

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×#3547	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6480 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=gh,f^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, 7534860);