#444401

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351597);