#197241

order	512
nilpotency class	3
isoclinism	isoclinic class of #196993
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×#3036	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1447 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104437);