#364273

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,d^2=g,e^2=i,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,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, 271469);