#7197860

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=hi,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105056);