#422609

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 329805);