#108654

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

506 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,e^c=ei,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15850);