#300412

order	512
nilpotency class	3
isoclinism	isoclinic class of #300397
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	#121	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

813 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)(3,8)(4,13)(5,14)(6,9)(7,10)(11,15)(12,16)(17,18,24,30)(19,31,35,42)(20,27,38,45)(21,39,37,26)(22,29,23,28)(25,34,43,33)(32,40,48,41)(36,46,47,44);(1,3,7,12)(2,8,10,16)(4,5,11,6)(9,13,14,15)(17,19,23,34,24,35,22,33)(18,25,29,42,30,43,28,31)(20,36,21,32,38,47,37,48)(26,44,45,40,39,46,27,41);(1,4,5,3,7,11,6,12)(2,8,9,13,10,16,14,15)(17,20,22,37,24,38,23,21)(18,26,28,27,30,39,29,45)(19,32,34,47,35,48,33,36)(25,40,42,46,43,41,31,44);(17,21,23,38,24,37,22,20)(18,27,29,26,30,45,28,39)(19,32,34,47,35,48,33,36)(25,41,42,44,43,40,31,46)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=eh,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,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207608);