#199238

order	512
nilpotency class	3
isoclinism	isoclinic class of #199230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1210 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)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18)(19,22,26,31)(20,23)(21,24)(25,28)(27,29,32,30);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(17,19)(18,22,24,31)(20,27)(21,26)(23,29,25,30)(28,32);(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(18,24)(19,26)(23,25)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249791, #72249798, #72249804, …
maximal normal subgroup of	#72249791, #72249798, #72249804, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106434);