#378334

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	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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)(9,10,14,21,15,20,32,19)(11,22,26,35,25,36,24,16)(12,27,28,38,29,17,30,39)(13,31,34,37,33,40,23,18);(9,11)(10,16)(12,13)(14,26)(15,25)(17,37)(18,27)(19,36)(20,35)(21,22)(23,30)(24,32)(28,34)(29,33)(31,38)(39,40);(1,3,5,4)(2,6,7,8)(9,12,15,29)(10,17,20,27)(11,23,25,34)(13,24,33,26)(14,28,32,30)(16,31,35,40)(18,22,37,36)(19,38,21,39);(1,4,5,3)(2,6,7,8)(9,13,32,23,15,33,14,34)(10,18,19,40,20,37,21,31)(11,12,24,30,25,29,26,28)(16,27,36,39,35,17,22,38)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=fh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei

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, 285530);