#120276

order	512
nilpotency class	3
isoclinism	isoclinic class of #10549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
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

556 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)(5,6)(7,10)(8,11)(9,12)(13,14,18,21)(15,22,20,17)(16,19,26,28)(23,25,27,24);(5,7,9,8)(6,10,12,11)(13,15,17,14,18,20,22,21)(16,19,24,27,26,28,25,23);(6,11)(7,8)(10,12)(13,16,17,24,18,26,22,25)(14,19,20,27,21,28,15,23)

Polycyclic presentation:

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

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, 27472);