#213024

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(2,4)(5,6,11,13,10,9,8,17)(7,12,14,20,16,15,18,19)(21,22,26,30)(23,31,35,45)(24,28,36,42)(25,38,32,29)(27,39,41,34)(33,44,47,52)(37,49,46,43)(40,50,51,48);(5,7)(6,12)(8,18)(9,15)(10,16)(11,14)(13,20)(17,19)(21,23,32,39,26,35,25,34)(22,27,38,45,30,41,29,31)(24,33,46,50,36,47,37,48)(28,40,49,52,42,51,43,44);(1,2,3,4)(5,8,10,11)(6,13,9,17)(7,14,16,18)(12,19,15,20)(21,24,26,36)(22,28,30,42)(23,33,35,47)(25,37,32,46)(27,40,41,51)(29,43,38,49)(31,44,45,52)(34,48,39,50);(5,9,11,17,10,6,8,13)(7,15,14,19,16,12,18,20)(24,36)(28,42)(33,47)(37,46)(40,51)(43,49)(44,52)(48,50)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fh,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120220);