#407980

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1246 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252628, #72256873, #72256881
maximal normal subgroup of	#72252628, #72256873, #72256881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315176);