#114542

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

885 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21738);