#300495

order	512
nilpotency class	3
isoclinism	isoclinic class of #300469
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	#19026	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

883 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:

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

Polycyclic presentation:

2^9|a^2=i,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=di,g^c=ghi

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, 207691);