#120680

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

498 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:

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

Polycyclic presentation:

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

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, 27876);