#425606

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

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, 332802);