#422228

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1008 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329424);