#112364

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

815 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)(7,14,16,11)(8,12)(9,13)(10,15,19,17)(18,20)(21,22)(23,31,34,27)(24,35)(25,38)(26,29)(28,37)(30,41)(32,48,49,52)(33,43,39,45)(36,46)(40,44)(42,50,47,51);(1,2)(3,4)(5,7)(6,10)(8,17)(9,16)(11,12)(13,19)(14,20)(15,18)(21,23,26,34)(22,27,29,31)(24,32,36,49)(25,39,41,33)(28,42,44,47)(30,45,38,43)(35,48,46,52)(37,51,40,50);(2,4)(5,7)(6,11)(8,15)(9,16)(10,12)(13,14)(17,18)(19,20)(21,24)(22,28)(23,32)(25,40)(26,36)(27,42)(29,44)(30,46)(31,47)(33,51)(34,49)(35,38)(37,41)(39,50)(43,48)(45,52)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19560);