#331006

order	512
nilpotency class	3
isoclinism	isoclinic class of #330975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

507 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238202);