#429606

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

987 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)(5,6,10,15,12,17,11,16)(7,18,21,32,22,33,30,13)(8,23,25,9,14,28,26,29)(19,27,34,35,24,36,31,20);(5,7,11,21,12,22,10,30)(6,13,16,18,17,32,15,33)(8,24,26,31,14,19,25,34)(9,27,23,35,29,36,28,20);(5,8,12,14)(6,9,17,29)(7,19,22,24)(10,26,11,25)(13,27,32,36)(15,23,16,28)(18,20,33,35)(21,31,30,34);(2,4)(5,9)(6,14)(7,20,22,35)(8,17)(10,28)(11,23)(12,29)(13,31,32,34)(15,26)(16,25)(18,19,33,24)(21,27,30,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=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, 336802);