#186413

order	512
nilpotency class	3
isoclinism	isoclinic class of #186137
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3506	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2384 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249711, #72249798, #72249810
maximal normal subgroup of	#72249711, #72249798, #72249810

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93609);