#194383

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60900	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,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, 101579);