#7156198

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61086	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

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, 7063394);