#405459

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312655);