#364152

order	512
nilpotency class	3
isoclinism	isoclinic class of #363904
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13960
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1387 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338483, #72338489, #72338498
maximal normal subgroup of	#72338483, #72338489, #72338498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271348);