#424599

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
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

1261 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei,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, 331795);