#300532

order	512
nilpotency class	3
isoclinism	isoclinic class of #300501
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	#18426	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

909 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=ehi,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=dhi,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, 207728);