#421007

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1840 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:

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

Polycyclic presentation:

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

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, 328203);