#184989

order	512
nilpotency class	3
isoclinism	isoclinic class of #184985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3460	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2375 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18,21,24,22,25,20,29)(19,23,28,31,26,30,27,32);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(1,3)(2,7)(4,11)(5,12)(6,14)(8,15)(9,10)(13,16)(19,26)(23,30)(27,28)(31,32);(17,20,22,21)(18,24,25,29)(19,27,26,28)(23,31,30,32)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ci,e^b=eh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92185);