#194315

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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 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)(3,10,13,16)(4,7)(5,14)(6,9)(8,12,11,15)(17,18,22,27,23,28,21,26)(19,24,30,38,31,39,36,42)(20,32,35,25,33,41,34,40)(29,43,45,48,44,47,46,37);(1,3,6,15,7,13,5,12)(2,8,9,16,4,11,14,10)(17,19,22,30,23,31,21,36)(18,24,27,38,28,39,26,42)(20,29,34,46,33,44,35,45)(25,37,32,47,40,48,41,43);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,32)(28,40)(30,46)(31,44)(36,45)(38,47)(39,48)(42,43);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,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, 101511);