#7626865

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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×#534	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6682 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:

(1,2,4,3)(6,12)(7,8)(10,14)(16,18)(21,22,26,25)(23,29,31,27)(24,28,32,33)(30,35,36,34);(5,6,11,12)(7,15,8,13)(9,14,20,10)(16,17,18,19)(21,22,26,25)(23,27,31,29)(24,28,32,33)(30,34,36,35);(5,7,11,8)(6,13,12,15)(9,16,20,18)(10,17,14,19)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,36)(33,35);(5,8,11,7)(6,13,12,15)(9,16,20,18)(10,19,14,17)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36);(1,3,4,2)(5,9)(6,14)(7,16)(8,18)(10,12)(11,20)(13,19)(15,17)(21,25,26,22)(23,29,31,27)(24,28,32,33)(30,34,36,35);(5,10,11,14)(6,9,12,20)(7,17,8,19)(13,16,15,18)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,36)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534061);