#114847

order	512
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60954	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22043);