#235190

order	512
nilpotency class	3
isoclinism	isoclinic class of #235187
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	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1390 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:

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

Polycyclic presentation:

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

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, 142386);