#456877

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1157 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,f^d=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, 364073);