#457564

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 364760);