#121262

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#397
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,3,4)(6,11)(7,12)(8,9)(13,14)(15,24)(16,21)(17,32)(18,23)(19,22)(20,26)(25,40)(27,41)(28,35)(29,34)(30,38)(31,39)(33,37)(36,42)(43,44);(5,6)(7,8)(9,11)(10,12)(13,15,19,28,18,27,17,26)(14,20,32,41,23,35,22,24)(16,29,30,42,31,43,33,25)(21,36,37,34,39,40,38,44);(5,7,9,12,10,11,8,6)(13,16,19,33,18,31,17,30)(14,21,32,38,23,39,22,37)(15,25,28,43,27,42,26,29)(20,34,41,36,35,44,24,40)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fi,e^c=ehi,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28458);