#116047

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#392
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

854 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)(6,9)(7,11)(10,12)(18,26)(21,28);(13,14)(15,21,22,28)(16,20)(17,19)(18,23,26,24)(25,27);(5,6,8,10)(7,12,11,9)(13,15)(14,18)(16,24)(17,22)(19,26)(20,28)(21,27)(23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215384, #72216614
maximal normal subgroup of	#72215384, #72216614
maximal quotient of	#72214224

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23243);