#195052

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 102248);