#379479

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286675);