#6628875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×2₋¹⁺⁶	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

16706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,f^b=fi,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262958, #72263043, #72267974, …
maximal normal subgroup of	#72262958, #72263043, #72267974, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536071);