#417565

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2362 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324761);