#377419

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284615);