#407964

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1258 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)(5,6,9,13,10,14,22,28)(7,15,18,23,17,27,21,11)(8,12,20,26,19,25,32,35)(16,29,31,34,30,36,33,24);(1,2,4,3)(8,19)(11,23)(12,25)(15,27)(16,30)(20,32)(26,35)(31,33);(5,7)(6,11)(8,16)(9,21)(10,17)(12,24)(13,27)(14,23)(15,28)(18,22)(19,30)(20,33)(25,34)(26,36)(29,35)(31,32);(1,3,4,2)(5,8,9,20,10,19,22,32)(6,12,13,26,14,25,28,35)(7,16,21,33,17,30,18,31)(11,24,27,36,23,34,15,29)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,f^c=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, 315160);