#476986

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1137 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)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,22,20,16,18,21)(15,17,25,28,24,26,23,27);(5,7)(6,9)(8,12)(10,11)(13,15,18,23,20,24,19,25)(14,17,21,27,16,26,22,28);(5,7)(6,10)(8,12)(9,11)(13,16,20,14)(15,17,24,26)(18,21,19,22)(23,28,25,27);(2,4)(7,12)(9,10)(13,17)(14,15)(16,24)(18,28)(19,27)(20,26)(21,25)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384182);