#300456

order	512
nilpotency class	3
isoclinism	isoclinic class of #300437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18428	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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)(3,11)(4,13)(5,9)(6,15)(7,10)(8,12)(14,16)(17,18,23,20)(19,27,31,42)(21,34,36,46)(22,33,37,26)(24,38,28,30)(25,40,32,35)(29,43,45,39)(41,47,48,44);(1,3,5,12)(2,8,9,11)(4,7,14,6)(10,13,15,16)(17,19,22,30,23,31,37,38)(18,24,26,27,20,28,33,42)(21,29,35,44,36,45,40,47)(25,39,34,41,32,43,46,48);(1,4,6,12,5,14,7,3)(2,8,10,13,9,11,15,16)(17,20,23,18)(19,28)(21,25,36,32)(22,33,37,26)(24,31)(27,38)(29,39)(30,42)(34,40,46,35)(41,44)(43,45)(47,48);(2,9)(8,11)(10,15)(13,16)(17,21,22,35,23,36,37,40)(18,25,26,34,20,32,33,46)(19,29,30,44,31,45,38,47)(24,39,27,41,28,43,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

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, 207652);