#458025

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

911 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:

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

Polycyclic presentation:

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

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, 365221);