#285531

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
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	#18566	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

694 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,f^c=fi,g^c=gh,e^d=eh

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, 192727);