#121661

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

504 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)(6,9)(7,11)(10,12)(13,14)(15,23)(16,28)(17,31)(18,22)(19,26)(20,30)(21,24)(25,41)(27,39)(29,36)(32,37)(33,42)(34,40)(35,44)(38,43);(5,6)(7,12)(8,10)(9,11)(13,15,24,40,18,27,17,26)(14,19,31,39,22,34,21,23)(16,25,29,42,30,43,32,44)(20,35,36,38,28,33,37,41);(5,6,8,10)(7,12,11,9)(13,16,17,32,18,30,24,29)(14,20,21,37,22,28,31,36)(15,25,26,44,27,43,40,42)(19,33,23,38,34,35,39,41)

Polycyclic presentation:

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

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, 28857);