#254316

order	512
nilpotency class	3
isoclinism	isoclinic class of #254299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

824 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:

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

Polycyclic presentation:

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

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, 161512);