#418481

order	512
nilpotency class	3
isoclinism	isoclinic class of #418480
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15263
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1679 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)(3,12)(4,16)(5,20)(6,22)(7,11)(8,14)(9,19)(10,21)(13,27)(15,26)(17,24)(18,25)(23,28)(29,32)(30,31);(1,3)(2,8)(4,13)(5,6)(7,15)(9,24)(10,26)(11,20)(12,22)(14,21)(16,23)(17,18)(19,29)(25,27)(28,31)(30,32);(1,4)(2,9)(3,13)(5,17)(6,18)(7,19)(8,23)(10,25)(11,16)(12,27)(14,28)(15,29)(20,24)(21,31)(22,30)(26,32);(1,5,21,15,7,14,6,3)(2,8,22,26,11,20,10,12)(4,17,31,29,19,28,18,13)(9,23,30,32,16,24,25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338544, #72338546, #72338561
maximal normal subgroup of	#72338544, #72338546, #72338561

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325677);