#496417

order	512
nilpotency class	4
isoclinism	isoclinic class of #480656
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,g^c=gh,g^f=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312277, #72312285, #72312302, …
maximal normal subgroup of	#72312277, #72312285, #72312302, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403613);