#115920

order	512
nilpotency class	3
isoclinism	isoclinic class of #99979
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

648 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,f^b=fh,e^c=ei,f^c=fh,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23116);