#285079

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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

621 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^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, 192275);