#185729

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#3483	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92925);