#7627643

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×#538	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6234 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)(21,22,26,29)(23,30,25,27)(24,33,34,28)(31,35,32,36);(7,14)(8,16)(11,13)(12,17)(21,23)(22,27)(24,31)(25,26)(28,35)(29,30)(32,34)(33,36);(6,10)(11,13)(12,17)(18,19)(21,24,26,34)(22,28,29,33)(23,31,25,32)(27,35,30,36);(5,7,9,14)(6,11,10,13)(8,15,16,20)(12,18,17,19)(21,24)(22,28)(23,32)(25,31)(26,34)(27,36)(29,33)(30,35);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,25,26,23)(22,27,29,30)(24,31,34,32)(28,36,33,35);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,22)(23,30)(24,28)(25,27)(26,29)(31,36)(32,35)(33,34)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gh,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, 7534839);