#207941

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

952 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)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,23)(22,24)(25,26)(27,30,34,39)(28,35)(29,32)(31,37)(33,40,36,38);(9,11,12,19,13,18,21,24)(10,14,16,17,15,22,20,23)(25,27)(26,30,32,39)(28,36)(29,34)(31,40,35,38)(33,37);(2,5)(6,8)(10,15)(11,18)(16,20)(19,24)(25,28)(26,31)(27,33)(29,37)(30,38)(32,35)(34,36)(39,40);(1,3,4,7)(2,6,5,8)(28,37)(31,35)(33,36)(38,40)

Polycyclic presentation:

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

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, 115137);