#6342582

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	Aut(G) order 524288
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	C₂²×2₋¹⁺⁴	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,6,5,7,3,8)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,31,34,26)(21,32,35,27)(23,28)(24,29)(25,30)(33,39,40,37)(36,38);(9,11,14,22)(10,15,18,19)(12,20,24,34)(13,21,25,35)(16,26,29,31)(17,27,30,32)(23,33,36,40)(28,37,38,39);(1,3,5,4)(2,6,7,8)(9,12,14,24)(10,16,18,29)(11,20,22,34)(13,23,25,36)(15,26,19,31)(17,28,30,38)(21,33,35,40)(27,37,32,39);(9,13)(10,17)(11,21)(12,23)(14,25)(15,27)(16,28)(18,30)(19,32)(20,33)(22,35)(24,36)(26,37)(29,38)(31,39)(34,40);(13,25)(17,30)(21,35)(23,36)(27,32)(28,38)(33,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,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.

Relationships

maximal subgroup of	#72281528, #72282071
maximal normal subgroup of	#72281528, #72282071

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249778);