#224279

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=hi,d^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131475);