#494879

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
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×#481
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

786 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)(7,15,16,32)(8,18)(9,22)(11,24,25,31)(13,28)(14,27)(17,33,23,20)(19,34)(21,35)(26,36,30,29);(1,2,4,3)(5,6)(7,11)(8,19)(9,14)(10,12)(13,21)(15,31)(16,25)(17,26)(18,34)(20,36)(22,27)(23,30)(24,32)(28,35)(29,33);(1,3,4,2)(5,7,9,23,10,16,22,17)(6,11,14,30,12,25,27,26)(8,20,18,15,21,33,35,32)(13,29,28,24,19,36,34,31);(6,12)(11,25)(13,19)(14,27)(24,31)(26,30)(28,34)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226790, #72226796, #72226797, …
maximal normal subgroup of	#72226790, #72226796, #72226797, …
maximal quotient of	#72285467, #72319652, #72342554

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402075);