#181359

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1722 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,b^a=be,c^a=ch,e^a=eh,g^a=gi,e^b=ei,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88555);