#443569

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#2021	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=eg,d^2=ei,b^a=be,c^a=cf,d^a=dh,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, 350765);