#438947

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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

808 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346143);