#388242

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295438);