#395555

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302751);