#388700

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=dei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295896);