#502148

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24291	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,7,10,23,31,30,14,19,12,18,13,17,32,22)(8,24,26,35,29,16,34,21,9,28,27,36,25,20,33,15);(1,2)(3,4)(5,7,11,23,10,30,31,19,14,18,12,17,13,22,32,6)(8,21,26,28,29,36,34,20,9,15,27,24,25,35,33,16);(2,4)(5,8,14,9)(6,15,19,21)(7,20,18,16)(10,25,13,29)(11,33,12,34)(17,35,23,36)(22,24,30,28)(26,31,27,32);(5,9,14,8)(6,16,19,20)(7,21,18,15)(10,25,13,29)(11,27,12,26)(17,24,23,28)(22,35,30,36)(31,33,32,34)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fh,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409344);