#192275

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 99471);