#7627619

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×#3542	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6956 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)(7,10)(8,11)(9,14)(12,17)(13,15)(16,19)(18,20)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(6,7)(8,15)(9,17)(18,19)(23,30)(24,32)(27,29)(28,33);(6,7)(8,15)(12,14)(16,20)(22,26)(27,29)(28,33)(34,35);(5,7,10,6)(8,13,15,11)(9,14,17,12)(16,19,20,18)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,36)(33,35);(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=i,d^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, 7534815);