#396095

order	512
nilpotency class	3
isoclinism	isoclinic class of #396094
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1583 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)(9,21)(10,14)(13,22)(16,23,30,25)(17,24,31,28)(19,32)(20,27)(26,33)(29,35,36,34);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,24)(14,15)(19,29)(20,30)(21,28)(22,31)(23,27)(26,35)(32,34)(33,36);(2,4)(5,8,10,20)(6,12,14,27)(7,16,18,30)(9,19,22,33)(11,23,15,25)(13,26,21,32)(17,29,31,36)(24,34,28,35);(5,9)(6,13,14,21)(7,17)(8,19)(10,22)(11,24,15,28)(12,26,27,32)(16,29)(18,31)(20,33)(23,34,25,35)(30,36)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,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	#72285165, #72285332, #72285421, …
maximal normal subgroup of	#72285165, #72285332, #72285421, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303291);