#120397

order	512
nilpotency class	3
isoclinism	isoclinic class of #10603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#396
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

(1,2)(3,4)(6,7)(8,9)(11,12)(13,14,17,22,19,23,18,21)(15,16,24,20,25,27,28,26);(5,6,9,11,10,12,8,7)(13,15,18,28,19,25,17,24)(14,20,22,27,23,26,21,16);(2,4)(5,7,8,12,10,11,9,6)(13,16,17,26,19,27,18,20)(14,15,21,24,23,25,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27593);