#419661

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1431 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:

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

Polycyclic presentation:

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

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, 326857);