#449676

order	512
nilpotency class	3
isoclinism	isoclinic class of #417558
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356872);