#147298

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,14,18,33)(4,21,23,55)(5,26)(6,11)(7,13)(8,16,34,17)(9,22,36,56)(10,29)(12,31)(15,32)(19,45,20,38)(24,39,25,58)(27,42,51,35)(28,37,57,43)(30,40)(41,59)(44,46)(47,54)(48,62)(49,64)(50,60)(52,61)(53,63);(1,3,6,19,7,18,31,20)(2,8,11,35,13,34,12,42)(4,15,24,48,23,49,25,53)(5,27,29,17,30,51,59,16)(9,32,37,62,36,64,43,63)(10,38,40,14,41,45,26,33)(21,54,39,61,55,44,58,60)(22,52,57,46,56,50,28,47);(1,4,6,24,7,23,31,25)(2,9,12,43,13,36,11,37)(3,15,19,48,18,49,20,53)(5,28,29,22,30,57,59,56)(8,32,42,63,34,64,35,62)(10,39,26,21,41,58,40,55)(14,44,38,61,33,54,45,60)(16,50,27,47,17,52,51,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307178, #72307219, #72307295, …
maximal normal subgroup of	#72307178, #72307219, #72307295, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54494);