#285106

order	512
nilpotency class	3
isoclinism	isoclinic class of #285051
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	#24170	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

660 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192302);