#214038

order	512
nilpotency class	3
isoclinism	isoclinic class of #214020
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121234);