#493435

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3402 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)(5,6)(7,17)(8,13)(9,14)(10,15,24,26)(11,16)(12,19,27,23)(18,28)(20,29,34,36)(21,30)(22,31)(25,32,35,33);(5,7,11,9)(6,12,16,27)(8,18,21,22)(10,23,24,19)(13,25,30,35)(14,26,17,15)(20,33,34,32)(28,29,31,36);(2,4)(5,8,11,21)(6,13,16,30)(7,18,9,22)(10,20,24,34)(12,25,27,35)(14,31,17,28)(15,29,26,36)(19,32,23,33);(5,9,11,7)(6,14,16,17)(8,18,21,22)(10,23,24,19)(12,26,27,15)(13,28,30,31)(20,32,34,33)(25,29,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302959, #72303028, #72303154, …
maximal normal subgroup of	#72302959, #72303028, #72303154, …
maximal quotient of	#72313526

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400631);