#235025

order	512
nilpotency class	3
isoclinism	isoclinic class of #235012
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×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1266 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:

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

Polycyclic presentation:

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

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, 142221);