#400693

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	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1443 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:

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

Polycyclic presentation:

2^9|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, 307889);