#381482

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288678);