#434063

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 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)(5,6,9,14,11,15,10,16)(7,12,19,26,18,28,20,27)(8,13,24,32,22,31,23,30)(17,33,35,29,34,25,21,36);(5,7)(6,12)(8,21,22,35)(9,19)(10,20)(11,18)(13,29,31,36)(14,26)(15,28)(16,27)(17,23,34,24)(25,30,33,32);(5,8)(6,13,15,31)(7,17,18,34)(9,23)(10,24)(11,22)(12,25)(14,30,16,32)(19,21,20,35)(26,29)(27,36)(28,33);(2,4)(6,14)(9,10)(12,26)(13,30)(15,16)(19,20)(21,35)(23,24)(25,29)(27,28)(31,32)(33,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi,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, 341259);