#108492

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#380
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

646 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:

(5,6,7,9,8,10,11,12)(13,14)(15,25)(16,30,32,37)(17,23)(18,22)(19,24)(20,27)(21,31,38,33)(26,34,41,44)(28,36)(29,35)(39,40,43,42);(1,2,3,4)(13,15,18,28,19,29,17,27)(14,20,23,35,24,36,22,25)(16,26,33,42,32,41,31,40)(21,34,30,43,38,44,37,39);(13,16,17,31,19,32,18,33)(14,21,22,37,24,38,23,30)(15,26,27,40,29,41,28,42)(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=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,f^c=fh

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, 15688);