#6342578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	Aut(G) order 1572864
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

4088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,g^b=gh,e^d=eh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249774);