#199399

order	512
nilpotency class	3
isoclinism	isoclinic class of #199230
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3038	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,4)(5,8)(6,7)(9,10,14,18,15,17,13,11)(12,19,20,16,21,24,22,23);(1,3,4,2,6,8,5,7)(9,11,14,17,15,18,13,10)(12,19,20,23,21,24,22,16);(9,12)(10,16)(11,19)(13,22)(14,20)(15,21)(17,23)(18,24);(1,4,6,5)(2,3,7,8)(10,17)(11,18)(16,23)(19,24)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=hi,e^2=hi,f^2=i,g^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106595);