#7627620

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×#3541	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6488 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)(21,22,27,31)(23,26,25,28)(24,30,35,29)(32,36,33,34);(7,14)(8,16)(11,13)(12,17)(21,23)(22,28)(24,32)(25,27)(26,31)(29,36)(30,34)(33,35);(6,10)(11,13)(12,17)(18,19)(21,24,27,35)(22,29,31,30)(23,32,25,33)(26,34,28,36);(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,25,27,23)(22,26,31,28)(24,32,35,33)(29,36,30,34);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,24)(22,30)(23,33)(25,32)(26,34)(27,35)(28,36)(29,31);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,26)(22,23)(24,34)(25,31)(27,28)(29,32)(30,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,c^2=g,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, 7534816);