#365041

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1662 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255889, #72255892, #72255916
maximal normal subgroup of	#72255889, #72255892, #72255916

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272237);