#405605

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1374 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,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, 312801);