#410802

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

681 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=h,c^2=fg,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317998);