#199971

order	512
nilpotency class	3
isoclinism	isoclinic class of #199824
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=i,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, 107167);