#414769

order	512
nilpotency class	3
isoclinism	isoclinic class of #394253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1498 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)(5,6,8,10)(7,11,12,9)(13,14,18,21)(15,20,16,26)(17,23,27,22)(19,25,28,24);(13,15,17,24,18,16,27,25)(14,19,22,20,21,28,23,26);(2,4)(13,16)(14,20)(15,18)(17,25)(19,23)(21,26)(22,28)(24,27);(5,7,8,12)(6,9,10,11)(14,21)(19,28)(20,26)(22,23)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321965);