#482650

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

920 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389846);