#343015

order	512
nilpotency class	3
isoclinism	isoclinic class of #343003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

695 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250211);