#395554

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3027	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2215 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 302750);