#6379030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342728
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂×#3489	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,10,16,11,15,9,14)(7,12,18,26,19,27,23,29)(8,20,21,30,22,13,24,28)(17,31,33,35,34,25,32,36);(1,2)(3,4)(5,7,11,19)(6,12,15,27)(8,17,22,34)(9,23,10,18)(13,25,20,31)(14,29,16,26)(21,33,24,32)(28,36,30,35);(5,8)(6,13)(7,17)(9,21)(10,24)(11,22)(12,25)(14,28)(15,20)(16,30)(18,32)(19,34)(23,33)(26,35)(27,31)(29,36);(2,4);(5,9,11,10)(6,14,15,16)(7,18,19,23)(8,21,22,24)(12,26,27,29)(13,28,20,30)(17,32,34,33)(25,35,31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,h^c=hi

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, 6286226);