#183703

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

636 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,g^2=h,b^a=be,d^a=dhi,c^b=ci,e^b=eh,f^b=fh,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, 90899);