#120752

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#388
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

570 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fi,g^b=gh,f^c=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27948);