#320124

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25599	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=eghi,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227320);