#461297

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 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:

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

Polycyclic presentation:

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

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, 368493);