#420143

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1259 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,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, 327339);