#106299

order	512
nilpotency class	3
isoclinism	isoclinic class of #9103
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci,e^b=eh,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13495);