#186815

order	512
nilpotency class	3
isoclinism	isoclinic class of #186808
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	#62814	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277922, #72278002, #72278163
maximal normal subgroup of	#72277922, #72278002, #72278163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94011);