#400694

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1403 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:

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

Polycyclic presentation:

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

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, 307890);