#495898

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

635 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320087, #72320189, #72320215, …
maximal normal subgroup of	#72320087, #72320189, #72320215, …
maximal quotient of	#72344017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403094);