#347914

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,5)(4,7)(6,8)(9,10,15,20)(11,21,26,46)(12,17,29,40)(13,18,32,43)(14,33,22,19)(16,34,37,25)(23,44,51,67)(24,45,30,58)(27,53,56,38)(28,57,47,39)(31,60,48,42)(35,61,63,50)(36,59,41,52)(49,69,54,66)(55,65,68,71)(62,70,64,72);(9,11,22,34,15,26,14,25)(10,16,33,46,20,37,19,21)(12,23,47,61,29,51,28,50)(13,30,48,52,32,24,31,59)(17,35,57,67,40,63,39,44)(18,41,60,45,43,36,42,58)(27,54,68,70,56,49,55,72)(38,64,71,66,53,62,65,69);(2,5)(3,8)(6,7)(9,12)(10,17)(11,23)(13,27)(14,28)(15,29)(16,35)(18,38)(19,39)(20,40)(21,44)(22,47)(24,49)(25,50)(26,51)(30,54)(31,55)(32,56)(33,57)(34,61)(36,62)(37,63)(41,64)(42,65)(43,53)(45,66)(46,67)(48,68)(52,70)(58,69)(59,72)(60,71);(9,13)(10,18)(11,24)(12,27)(14,31)(15,32)(16,36)(17,38)(19,42)(20,43)(21,45)(22,48)(23,49)(25,52)(26,30)(28,55)(29,56)(33,60)(34,59)(35,62)(37,41)(39,65)(40,53)(44,66)(46,58)(47,68)(50,70)(51,54)(57,71)(61,72)(63,64)(67,69)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,f^c=fi,g^c=gh,f^d=fh

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, 255110);