#185561

order	512
nilpotency class	3
isoclinism	isoclinic class of #185475
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	C₂²×#553	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240220, #72240221, #72240243, …
maximal normal subgroup of	#72240220, #72240221, #72240243, …
maximal quotient of	#72240287

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92757);