#426744

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1051 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322821
maximal normal subgroup of	#72322821

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333940);