#116552

order	512
nilpotency class	3
isoclinism	isoclinic class of #98904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=i,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^c=ehi,g^c=gh

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, 23748);