#376938

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284134);