#182998

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

878 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,e^a=eh,g^a=ghi,e^b=ei,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90194);