#211725

order	512
nilpotency class	3
isoclinism	isoclinic class of #19230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂³	#525
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118921);