#120556

order	512
nilpotency class	3
isoclinism	isoclinic class of #120555
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,9,11,16)(4,7)(5,8)(6,10,14,13)(12,15)(17,18,22,25,23,27,21,26)(19,20,29,31,30,32,28,24);(4,12)(6,14)(7,15)(10,13)(17,19,21,28,23,30,22,29)(18,24,25,20,27,31,26,32);(1,3)(2,6)(4,13)(5,11)(7,16)(8,14)(9,15)(10,12)(17,20,21,31,23,32,22,24)(18,19,25,28,27,30,26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27752);