#214275

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#34964	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1041 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^b=dhi,d^c=di,f^c=fh

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, 121471);