#111565

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₈	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1288 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)(5,6,7,9,8,10,11,12)(13,14)(15,21,24,18)(16,20)(17,19)(22,27,23,26)(25,28);(13,15)(14,18)(16,22)(17,24)(19,21)(20,26)(23,25)(27,28);(2,4)(15,22)(16,25)(18,26)(20,28)(21,27)(23,24)

Polycyclic presentation:

2^9|a^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fh,g^b=gh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237149, #72237150, #72237152
maximal normal subgroup of	#72237149, #72237150, #72237152

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18761);