#288935

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25392	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=eh,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, 196131);