#491140

order	512
nilpotency class	4
isoclinism	isoclinic class of #491097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=egi,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285888
maximal normal subgroup of	#72285888

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398336);