#112300

order	512
nilpotency class	3
isoclinism	isoclinic class of #97710
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#37547	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

917 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19496);