#285122

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

613 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)(3,8)(4,12)(5,7)(6,10)(9,11)(13,15)(14,16)(17,18,22,24,21,25,20,30)(19,26,29,32,28,23,27,31);(2,7)(3,5)(8,10)(9,12)(11,14)(15,16)(17,19,22,29,21,28,20,27)(18,23,24,31,25,26,30,32);(1,3,6,5)(2,8,10,7)(4,11,13,14)(9,15,16,12)(17,18)(19,26)(20,24)(21,25)(22,30)(23,28)(27,32)(29,31);(1,4,6,13)(2,9,10,16)(3,11,5,14)(7,12,8,15)(17,20,21,22)(18,24,25,30)(19,27,28,29)(23,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192318);