#494324

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
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

1052 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)(6,12,14,29)(7,17,19,22)(8,9)(10,24)(13,30)(15,26,21,16)(18,20,35,27)(23,25)(28,36)(31,33);(2,4)(5,6,10,26,11,14,24,16)(7,13,27,33,19,31,20,30)(8,21,9,12,23,15,25,29)(17,34,35,36,22,32,18,28);(5,7)(6,13)(8,22)(9,18)(10,27)(11,19)(12,28)(14,31)(15,34)(16,30)(17,23)(20,24)(21,32)(25,35)(26,33)(29,36);(6,14)(12,29)(13,31)(15,21)(16,26)(28,36)(30,33)(32,34)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eh,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	#72249864, #72249871, #72249885, …
maximal normal subgroup of	#72249864, #72249871, #72249885, …
maximal quotient of	#72265681, #72279601, #72285353

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401520);