#493984

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
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×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292959, #72292960, #72292964, …
maximal normal subgroup of	#72292959, #72292960, #72292964, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401180);