#400649

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
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

860 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)(5,6,9,12)(7,11,8,10)(13,14,19,22)(15,18,21,17)(16,23,27,28)(20,26,24,25);(5,7,9,8)(6,10,12,11)(13,15,18,22,19,21,17,14)(16,24,25,28,27,20,26,23);(5,7,9,8)(6,11,12,10)(13,16,17,26,19,27,18,25)(14,20,21,28,22,24,15,23);(2,4)(5,8,9,7)(6,10,12,11)(13,17,19,18)(14,21,22,15)(16,25,27,26)(20,23,24,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,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, 307845);