#376724

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1307 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)(5,6,12,17,10,16,11,32)(7,18,22,34,21,13,33,35)(8,23,27,36,25,20,26,14)(9,28,31,19,29,15,30,24);(2,4)(5,7)(6,13)(8,24)(9,20)(10,21)(11,33)(12,22)(14,31)(15,26)(16,18)(17,35)(19,25)(23,29)(27,28)(30,36)(32,34);(5,8,11,26,10,25,12,27)(6,14,17,23,16,36,32,20)(7,19,22,15,21,24,33,28)(9,18,30,35,29,13,31,34);(2,4)(5,9,12,31,10,29,11,30)(6,15,17,24,16,28,32,19)(7,20,22,14,21,23,33,36)(8,18,27,34,25,13,26,35)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283920);