#398916

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
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

981 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306112);