#112094

order	512
nilpotency class	3
isoclinism	isoclinic class of #99947
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

916 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)(5,6,10,7)(8,11,9,12)(13,14,18,24,19,22,17,23)(15,25,28,35,29,20,27,36)(16,21,31,39,32,38,33,40)(26,41,30,44,42,34,43,37);(6,11)(7,12)(8,9)(13,15,19,29)(14,20,22,25)(16,30,32,43)(17,28,18,27)(21,37,38,44)(23,36,24,35)(26,33,42,31)(34,40,41,39);(2,4)(5,7,10,6)(8,12,9,11)(13,16,19,32)(14,21,22,38)(15,26,29,42)(17,33,18,31)(20,34,25,41)(23,40,24,39)(27,43,28,30)(35,44,36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19290);