#111668

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#379
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6,9,12,10,7,8,11)(13,14,17,22,18,21,32,36)(15,23,26,37,25,19,30,34)(16,27,29,38,28,20,31,35)(24,39,41,44,40,33,42,43);(1,2)(3,4)(5,7,10,6)(8,11,9,12)(13,15)(14,19)(16,24)(17,30)(18,25)(20,33)(21,23)(22,37)(26,32)(27,39)(28,40)(29,42)(31,41)(34,36)(35,44)(38,43);(2,4)(13,16)(14,20)(15,24)(17,31)(18,28)(19,33)(21,27)(22,38)(23,39)(25,40)(26,42)(29,32)(30,41)(34,44)(35,36)(37,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18864);