#7622580

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	C₂×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,g^2=h,h^2=i,b^a=bf,d^a=di,e^a=ei,e^b=ei,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315166, #72315168, #72315170, …
maximal normal subgroup of	#72315166, #72315168, #72315170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529776);