#375837

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283033);