#410692

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

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

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, 317888);