#376719

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
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

1301 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)(7,14,8,22,20,13,21,30)(9,24,28,33,26,15,27,32)(19,29,23,36,34,31,25,35);(2,4)(5,7)(6,13)(8,12)(9,19)(10,20)(11,21)(14,16)(15,29)(17,30)(18,22)(23,28)(24,31)(25,27)(26,34)(32,36)(33,35);(5,8,11,7,10,21,12,20)(6,14,17,22,16,13,18,30)(9,25,27,34,26,23,28,19)(15,29,32,36,24,31,33,35);(2,4)(5,9,12,28,10,26,11,27)(6,15,17,32,16,24,18,33)(7,19,8,23,20,34,21,25)(13,29,30,36,14,31,22,35)

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,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250492, #72252977, #72256886
maximal normal subgroup of	#72250492, #72252977, #72256886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283915);