#434240

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 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)(5,6,11,16,10,17,12,15)(7,13,19,30,9,27,20,29)(8,21,25,14,24,32,26,33)(18,34,35,28,23,31,22,36);(5,7)(6,13)(8,22)(9,10)(11,20)(12,19)(14,31)(15,30)(16,29)(17,27)(18,26)(21,36)(23,25)(24,35)(28,32)(33,34);(1,2,3,4)(5,8,11,25,10,24,12,26)(6,14,15,21,17,33,16,32)(7,18,19,35,9,23,20,22)(13,28,29,34,27,36,30,31);(5,9,10,7)(6,13)(8,23,24,18)(11,20,12,19)(14,28)(15,29)(16,30)(17,27)(21,34)(22,26,35,25)(31,32)(33,36)

Polycyclic presentation:

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

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, 341436);