#430459

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1247 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287080, #72287334, #72287360
maximal normal subgroup of	#72287080, #72287334, #72287360

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337655);