#235514

order	512
nilpotency class	3
isoclinism	isoclinic class of #235504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142710);