#376718

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
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

1315 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)(5,6,12,17,10,16,11,18)(7,14,8,22,20,13,21,29)(9,24,27,32,25,15,26,31)(19,28,23,36,33,30,34,35);(2,4)(5,7)(6,13)(8,12)(9,19)(10,20)(11,21)(14,16)(15,28)(17,29)(18,22)(23,27)(24,30)(25,33)(26,34)(31,36)(32,35);(5,8,11,7,10,21,12,20)(6,14,17,22,16,13,18,29)(9,23,26,19,25,34,27,33)(15,30,31,35,24,28,32,36);(2,4)(5,9,12,27,10,25,11,26)(6,15,17,31,16,24,18,32)(7,19,8,23,20,33,21,34)(13,28,29,36,14,30,22,35)

Polycyclic presentation:

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

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, 283914);