#111846

order	512
nilpotency class	3
isoclinism	isoclinic class of #9996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#3036	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1312 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)(5,6,8,10,9,12,7,11)(13,14,18,21)(15,23)(16,20,17,22)(19,26)(24,28)(25,27);(2,4)(6,10)(7,8)(11,12)(13,15)(14,19)(16,25)(17,24)(18,26)(20,28)(21,23)(22,27);(14,20)(15,24)(16,17)(21,22)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257799, #72257884, #72257885, …
maximal normal subgroup of	#72257799, #72257884, #72257885, …
maximal quotient of	#72264805, #72265025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19042);