#343089

order	512
nilpotency class	3
isoclinism	isoclinic class of #343075
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

672 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defhi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250285);