#497368

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404564);