#208453

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=hi,d^2=f,e^2=hi,g^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, 115649);