#105258

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#376
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

742 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)(5,6,7,10,8,9,11,12)(13,14)(15,25,28,35)(16,21)(17,22)(18,24)(19,23)(20,27,34,29)(26,39,41,44)(30,36)(31,38)(32,37)(33,40,43,42);(2,4)(13,15)(14,20,23,34)(16,26)(17,27)(18,29)(19,28)(21,33,37,43)(22,25,24,35)(30,40)(31,42)(32,41)(36,39,38,44);(6,9)(10,12)(13,16,18,31,19,32,17,30)(14,21,24,38,23,37,22,36)(15,26,29,42,28,41,27,40)(20,33,35,44,34,43,25,39)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ci,f^a=fh,e^b=eh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12454);