#494707

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	#35676	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2054 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)(5,6,9,15,10,14,11,23)(7,12,17,28,18,27,24,32)(8,19,21,33,22,13,25,30)(16,29,31,36,20,26,34,35);(5,6,11,15)(7,12,24,28)(8,13,25,30)(9,23,10,14)(16,26,34,35)(17,32,18,27)(19,21,33,22)(20,29,31,36);(2,4)(5,7,10,18)(6,12,14,27)(8,20,22,16)(9,17,11,24)(13,29,19,26)(15,28,23,32)(21,34,25,31)(30,36,33,35);(5,8,10,22)(6,13,14,19)(7,16,18,20)(9,21,11,25)(12,26,27,29)(15,30,23,33)(17,31,24,34)(28,35,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317729, #72318728, #72318734
maximal normal subgroup of	#72317729, #72318728, #72318734

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401903);