#205721

order	512
nilpotency class	3
isoclinism	isoclinic class of #205565
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×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1406 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)(3,7)(4,8)(5,6)(9,10,14,18,15,17,13,11)(12,20,23,24,21,19,22,16);(1,3,7,2,5,8,4,6)(9,11,14,17,15,18,13,10)(12,20,23,16,21,19,22,24);(2,6)(3,8)(9,12,14,23,15,21,13,22)(10,16,11,19,17,24,18,20);(12,21)(16,24)(19,20)(22,23)

Polycyclic presentation:

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

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, 112917);