#195040

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
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

1150 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:

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

Polycyclic presentation:

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

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, 102236);