#332874

order	512
nilpotency class	3
isoclinism	isoclinic class of #332839
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

684 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240070);