#6692734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257583, #72257656, #72264324
maximal normal subgroup of	#72257583, #72257656, #72264324
maximal quotient of	#72265136

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6599930);