#419824

order	512
nilpotency class	3
isoclinism	isoclinic class of #419466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251368, #72251395, #72251403, …
maximal normal subgroup of	#72251368, #72251395, #72251403, …
maximal quotient of	#72251443

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327020);