#494887

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

790 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250341, #72250344, #72250376, …
maximal normal subgroup of	#72250341, #72250344, #72250376, …
maximal quotient of	#72319126, #72320019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402083);