#494270

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249864, #72249872, #72249879, …
maximal normal subgroup of	#72249864, #72249872, #72249879, …
maximal quotient of	#72280706, #72285235, #72308328

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401466);