#207924

order	512
nilpotency class	3
isoclinism	isoclinic class of #162486
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

990 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)(3,6)(4,5)(7,8)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24)(25,26)(27,30,34,37)(28,31)(29,32)(33,36,39,40)(35,38);(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24)(25,27)(26,30,32,37)(28,33)(29,34)(31,36,38,40)(35,39);(2,5)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3,4,7)(2,6,5,8)(11,17)(15,22)(18,23)(20,24)(26,32)(28,35)(30,37)(33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh

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, 115120);