#448833

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=i,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356029);