#331004

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,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, 238200);