#499210

order	512
nilpotency class	4
isoclinism	isoclinic class of #499210
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2058
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249773, #72249781, #72249835, …
maximal normal subgroup of	#72249773, #72249781, #72249835, …
maximal quotient of	#72256841, #72322921

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406406);