#322270

order	512
nilpotency class	3
isoclinism	isoclinic class of #288931
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

801 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)(5,6,10,7)(8,9,12,11)(13,14,16,22,18,21,17,20)(15,19,23,27,25,26,24,28);(5,7,9,12,10,6,11,8)(13,15,16,23,18,25,17,24)(14,19,22,27,21,26,20,28);(2,4)(14,20)(15,23)(16,17)(21,22)(24,25)(27,28);(5,8,9,7,10,12,11,6)(14,21)(19,26)(20,22)(27,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229466);