#496426

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
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×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,27)(20,31)(21,26)(22,36,35,24)(23,28)(25,33)(29,38)(30,39,42,37)(32,41)(34,40,46,44)(43,47,45,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,28)(18,23,26,27)(20,32,33,29)(22,30,35,42)(24,39,36,37)(25,41,31,38)(34,45,46,43)(40,47,44,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,28)(23,37)(25,40)(26,36)(27,39)(29,32)(30,42)(31,44)(38,47)(41,48)(43,45);(2,8)(6,9)(7,16)(14,15)(17,20,21,33)(18,25,26,31)(19,29,28,32)(22,34,35,46)(23,38,27,41)(24,40,36,44)(30,43,42,45)(37,47,39,48)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403622);