#458567

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 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)(13,24)(16,29,32,36)(18,30)(20,27,23,28)(25,31,35,33)(26,34);(1,2)(3,4)(5,7,9,17,10,18,24,34)(6,11,13,15,14,26,22,30)(8,16,23,33,21,32,20,31)(12,25,28,29,19,35,27,36);(2,4)(8,20,21,23)(12,27,19,28)(16,31,32,33)(25,36,35,29);(5,8)(6,12)(7,16)(9,23)(10,21)(11,25)(13,28)(14,19)(15,29)(17,33)(18,32)(20,24)(22,27)(26,35)(30,36)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365763);