#375082

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2366
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

890 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)(5,6,9,12,11,16,10,15)(7,13,17,25,19,27,24,29)(8,20,22,14,21,26,23,30)(18,32,31,28,34,35,33,36);(6,12,16,15)(7,17,19,24)(8,21)(13,27)(14,20,30,26)(18,33,34,31)(22,23)(25,29);(2,4)(5,7,9,17,11,19,10,24)(6,13,12,25,16,27,15,29)(8,18,22,31,21,34,23,33)(14,28,26,35,30,36,20,32);(5,8,9,22,11,21,10,23)(6,14,15,20,16,30,12,26)(7,18,17,31,19,34,24,33)(13,28,29,32,27,36,25,35)

Polycyclic presentation:

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

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, 282278);