#455827

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1095 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,e^d=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, 363023);