#230414

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

549 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137610);