#427102

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

864 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)(5,6,9,11)(7,10,8,12)(13,14,19,16,18,22,17,21)(15,23,27,28,24,20,26,25);(5,7,9,8)(6,10,11,12)(14,16)(15,24)(17,19)(20,28)(21,22)(23,25);(5,8,9,7)(6,10,11,12)(13,15,19,27,18,24,17,26)(14,20,16,25,22,23,21,28);(2,4)(13,16,17,14,18,21,19,22)(15,25,26,20,24,28,27,23)

Polycyclic presentation:

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

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, 334298);