#416610

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3151	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1856 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=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, 323806);