#493429

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3487	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3386 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)(10,24,25,15)(11,16)(12,19,27,23)(18,28)(20,33,34,29)(21,30)(22,31)(26,32,36,35);(5,7,11,9)(6,12,16,27)(8,18,21,22)(10,19,25,23)(13,26,30,36)(14,15,17,24)(20,32,34,35)(28,33,31,29);(2,4)(5,8,11,21)(6,13,16,30)(7,18,9,22)(10,20,25,34)(12,26,27,36)(14,31,17,28)(15,29,24,33)(19,32,23,35);(5,9,11,7)(6,14,16,17)(8,18,21,22)(10,23,25,19)(12,15,27,24)(13,28,30,31)(20,32,34,35)(26,33,36,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303210, #72303216, #72303441, …
maximal normal subgroup of	#72303210, #72303216, #72303441, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400625);