#418569

order	512
nilpotency class	3
isoclinism	isoclinic class of #417454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1548 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)(6,9)(10,11)(13,14)(15,20)(16,25)(17,22)(18,27)(19,23)(21,26)(24,28);(5,6,8,9)(7,11,12,10)(13,15)(14,20)(16,19)(17,24)(18,26)(21,27)(22,28)(23,25);(13,16,19,15)(14,21,23,28)(17,26,18,24)(20,27,25,22);(2,4)(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,23,27)(15,24,16,26)(20,28,25,21)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=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, 325765);