#333089

order	512
nilpotency class	3
isoclinism	isoclinic class of #333055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240285);