#193769

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1045 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,e^b=eh,g^b=gh,d^c=di,f^c=fh

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, 100965);