#199827

order	512
nilpotency class	3
isoclinism	isoclinic class of #199824
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107023);