#238808

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60900	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=gi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308275, #72308389, #72308396
maximal normal subgroup of	#72308275, #72308389, #72308396

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146004);