#110085

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

786 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)(5,6,10,12,9,8,11,7)(13,14)(15,19,24,31)(16,20,26,33)(17,28)(18,21)(22,29)(23,35)(25,39,38,32)(27,41,40,34)(30,37)(36,42)(43,44);(2,4)(5,7,11,8,9,12,10,6)(13,15,17,25,18,24,29,38)(14,19,22,32,21,31,28,39)(16,23,27,36,26,37,40,44)(20,30,34,42,33,35,41,43);(5,8,11,12,9,6,10,7)(13,16)(14,20,21,33)(15,23)(17,27)(18,26)(19,30,31,35)(22,34,28,41)(24,37)(25,36)(29,40)(32,42,39,43)(38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ci,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=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, 17281);