#108397

order	512
nilpotency class	2
isoclinism	isoclinic class of #94871
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#376
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 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)(5,6,9,12,8,11,10,7)(13,14)(15,23,26,36)(16,20)(17,21)(18,22)(19,25,31,39)(24,35,38,43)(27,32)(28,33)(29,34)(30,37,41,44)(40,42);(5,7,10,11,8,12,9,6)(13,15)(14,19,22,31)(16,24)(17,25)(18,26)(20,30,33,41)(21,23,34,36)(27,37)(28,38)(29,39)(32,35,42,43)(40,44);(2,4)(6,11)(7,12)(13,16,17,27,18,28,29,40)(14,20,21,32,22,33,34,42)(15,24,25,37,26,38,39,44)(19,30,23,35,31,41,36,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,g^a=gh,c^b=ci,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15593);