#494043

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313447, #72313451, #72313481, …
maximal normal subgroup of	#72313447, #72313451, #72313481, …
maximal quotient of	#72343691

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401239);