#400321

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

983 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)(7,11)(8,10)(9,12)(13,14,20,26)(15,27,30,22)(16,21,29,37)(17,32,34,40)(18,25,19,24)(23,35,39,33)(28,42,43,36)(31,41,44,38);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,19,29,20,30,18,16)(14,21,25,22,26,37,24,27)(17,28,35,44,34,43,33,31)(23,36,40,38,39,42,32,41);(2,4)(5,7,9,8)(6,11,12,10)(13,16,18,30,20,29,19,15)(14,22,24,21,26,27,25,37)(17,31,33,43,34,44,35,28)(23,38,32,36,39,41,40,42);(5,8,9,7)(6,10,12,11)(13,17,20,34)(14,23,26,39)(15,28,30,43)(16,31,29,44)(18,33,19,35)(21,36,37,42)(22,38,27,41)(24,32,25,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307517);