#6379013

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	C₂⁴	C₂⁵
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4744 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)(7,11)(8,18,20,12)(9,13)(10,14)(15,29,31,23)(16,24)(17,25)(19,32,34,26)(21,27)(22,28)(30,36,33,35);(1,2)(3,4)(5,7,10,17)(6,11,14,25)(8,15,20,31)(9,21,22,16)(12,23,18,29)(13,27,28,24)(19,33,34,30)(26,36,32,35);(5,8)(6,12)(7,15)(9,19)(10,20)(11,23)(13,26)(14,18)(16,30)(17,31)(21,33)(22,34)(24,35)(25,29)(27,36)(28,32);(2,4);(5,9,10,22)(6,13,14,28)(7,16,17,21)(8,19,20,34)(11,24,25,27)(12,26,18,32)(15,30,31,33)(23,35,29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^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.

Relationships

maximal subgroup of	#72281079, #72283147, #72283331
maximal normal subgroup of	#72281079, #72283147, #72283331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286209);