#378180

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
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

1392 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)(11,21)(12,24)(13,28)(14,31)(15,20)(16,23)(17,27)(18,30)(19,29)(22,38)(25,33)(26,36)(32,35)(34,40)(37,39);(1,2)(3,4)(5,7,8,6)(9,11,14,30,15,23,29,13)(10,16,19,18,20,21,31,28)(12,22,32,39,26,40,27,25)(17,33,36,34,35,37,24,38);(2,4)(6,7)(9,12)(10,17)(11,22)(13,25)(14,32)(15,26)(16,33)(18,34)(19,36)(20,35)(21,37)(23,40)(24,31)(27,29)(28,38)(30,39);(9,13,29,23,15,30,14,11)(10,18,31,16,20,28,19,21)(12,25,27,40,26,39,32,22)(17,34,24,33,35,38,36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=efhi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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, 285376);