#120672

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#393
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

608 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:

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

Polycyclic presentation:

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

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, 27868);