#481653

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=hi,f^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,f^c=fi,f^e=fi

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, 388849);