#419608

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
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	C₂⁵	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1729 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)(15,18)(16,23,24,19)(17,20)(21,25,27,28)(22,26);(1,2)(3,4)(5,7,9,12)(6,10,8,11);(2,4)(5,8,9,6)(7,11,12,10)(13,15,17,22)(14,18,20,26)(16,21,24,27)(19,25,23,28);(7,12)(10,11)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,27)(26,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,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, 326804);