#230407

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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

553 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)(3,8,12,14,13,16,11,15)(17,18,24,28)(19,29)(20,21,36,39)(22,27,23,34)(25,40)(26,41,38,35)(30,45)(31,47)(32,44)(33,43)(37,42)(46,48);(2,7)(3,11)(4,5)(9,10)(12,13)(14,15)(17,19,23,32,24,33,22,40)(18,25,34,29,28,44,27,43)(20,30,41,46,36,47,35,42)(21,37,38,45,39,48,26,31);(17,20)(18,26)(19,30)(21,34)(22,35)(23,41)(24,36)(25,31)(27,39)(28,38)(29,37)(32,46)(33,47)(40,42)(43,48)(44,45);(1,3,5,12,6,13,4,11)(2,8,7,14,10,16,9,15)(17,21,23,38,24,39,22,26)(18,20,34,41,28,36,27,35)(19,31,40,48,33,45,32,37)(25,42,43,47,44,46,29,30)

Polycyclic presentation:

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

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, 137603);