#494535

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226790, #72226794, #72226796, …
maximal normal subgroup of	#72226790, #72226794, #72226796, …
maximal quotient of	#72285533, #72319677, #72342565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401731);