#377319

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=h,e^2=i,f^2=i,g^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,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

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, 284515);