#367800

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

838 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)(9,10,14,21,16,22,15,20)(11,19,24,17,25,33,13,31)(12,26,28,36,30,35,29,18)(23,32,37,40,38,39,27,34);(1,3,5,4)(2,6,7,8)(9,11,15,13,16,25,14,24)(10,17,21,33,22,31,20,19)(12,27,28,23,30,37,29,38)(18,34,35,39,36,40,26,32);(1,4,5,3)(2,6,7,8)(9,12,14,28,16,30,15,29)(10,18,20,35,22,36,21,26)(11,23,24,37,25,38,13,27)(17,32,19,34,31,39,33,40);(2,7)(6,8)(9,13,15,25,16,24,14,11)(10,19,21,17,22,33,20,31)(12,27,29,38,30,37,28,23)(18,34,26,32,36,40,35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=e,d^2=f,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,d^b=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274996);