#347497

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63069	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268465
maximal normal subgroup of	#72268465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254693);