#285230

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

627 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)(3,10)(4,6)(5,7)(8,11)(9,13)(12,16)(14,15)(17,18,21,25,22,20,27,24)(19,26,30,28,31,32,29,23);(2,6)(4,13)(7,9)(8,10)(11,14)(15,16)(17,19,27,29,22,31,21,30)(18,23,24,32,20,28,25,26);(2,7)(6,9)(8,15)(10,16)(17,20)(18,22)(19,28)(21,25)(23,31)(24,27)(26,30)(29,32);(1,3,5,12)(2,8,7,15)(4,14,13,11)(6,10,9,16)(17,18)(19,23)(20,22)(21,25)(24,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,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, 192426);