#413510

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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:

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

Polycyclic presentation:

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

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, 320706);