#7622637

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622636
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#442
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#62636	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #2875 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,f^2=h,h^2=i,b^a=bf,c^a=ch,f^a=fh,h^a=hi,f^b=fhi,g^b=gi,h^b=hi,f^c=fi,e^d=ei

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213937, #72213945, #72214039, …
maximal normal subgroup of	#72213937, #72213945, #72214039, …
maximal quotient of	#72216441, #72216726, #72216730, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529833);