#7627670

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×#3566	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6408 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)(5,6,10,8)(7,13,14,11)(9,12,17,16)(15,19,20,18)(21,22)(23,30)(24,33)(25,28)(26,27)(29,34)(31,35)(32,36);(7,14)(9,17)(11,13)(12,16)(21,23)(22,28)(24,31)(25,27)(26,30)(29,35)(32,34)(33,36);(6,8)(11,13)(12,16)(18,19)(21,24,27,34)(22,29,26,33)(23,31,25,32)(28,35,30,36);(5,7,10,14)(6,11,8,13)(9,15,17,20)(12,18,16,19)(21,24)(22,29)(23,32)(25,31)(26,33)(27,34)(28,36)(30,35);(5,8,10,6)(7,13,14,11)(9,16,17,12)(15,19,20,18)(21,25,27,23)(22,28,26,30)(24,31,34,32)(29,36,33,35);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,26,27,22)(23,28,25,30)(24,33,34,29)(31,35,32,36)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gh,e^2=h,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, 7534866);