#424010

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

989 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)(5,6,9,8)(7,11,12,10)(13,14)(15,24)(16,19)(17,21)(18,22)(20,26)(23,28)(25,27);(5,7,9,12)(6,10,8,11)(13,14)(15,25)(16,20)(17,21)(18,23)(19,26)(22,28)(24,27);(1,2,3,4)(13,15)(14,19)(16,17)(18,27)(20,22)(21,24)(23,25)(26,28);(5,8,9,6)(7,10,12,11)(13,16)(14,20)(15,17)(18,27)(19,22)(21,25)(23,24)(26,28)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fi,d^2=e,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,g^c=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, 331206);