#7627681

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=g,f^2=gh,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, 7534877);