#190647

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97843);