#7627677

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×#3549	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7278 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)(5,6,9,10)(7,11,13,17)(8,15,16,12)(14,20,19,18)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(6,10)(7,13)(8,16)(18,20)(23,30)(24,32)(27,29)(28,33);(6,10)(7,13)(12,15)(14,19)(22,26)(27,29)(28,33)(34,35);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,23,25,30)(22,27,26,29)(24,31,32,36)(28,34,33,35);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(5,8,9,16)(6,12,10,15)(7,14,13,19)(11,18,17,20)(21,24,25,32)(22,28,26,33)(23,31,30,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,d^2=gh,f^2=h,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, 7534873);