#404039

order	512
nilpotency class	3
isoclinism	isoclinic class of #396094
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284233, #72284459, #72285042, …
maximal normal subgroup of	#72284233, #72284459, #72285042, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311235);