#300525

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 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)(19,27)(20,31)(21,35)(22,26)(23,29)(24,33)(25,36)(28,42)(30,41)(32,45)(34,46)(37,44)(38,39)(40,43)(47,48);(1,3,5,11)(2,7,8,10)(4,14,9,16)(6,12,15,13)(17,19,21,29,22,30,36,39)(18,23,25,27,26,38,35,41)(20,28,33,43,34,44,45,47)(24,37,31,40,32,42,46,48);(1,4,6,8,5,9,15,2)(3,7,12,14,11,10,13,16)(17,18)(19,23,30,38)(20,32)(21,25)(22,26)(24,34)(27,39,41,29)(28,42,44,37)(31,45)(33,46)(35,36)(40,43,48,47);(2,8)(4,9)(7,10)(14,16)(17,20,21,33,22,34,36,45)(18,24,25,31,26,32,35,46)(19,28,29,43,30,44,39,47)(23,37,27,40,38,42,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,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=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, 207721);