#185361

order	512
nilpotency class	3
isoclinism	isoclinic class of #185273
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62819	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2321 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)(3,9,11,6)(4,7)(5,8)(10,13,16,15)(12,14)(17,18,22,26,23,19,21,25)(20,24,30,32,28,27,29,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,23,18)(20,27,28,24)(21,25,22,26)(29,31,30,32);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,31)(26,32);(2,8)(6,9)(7,14)(13,15)(20,28)(24,27)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92557);