#455825

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

1077 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:

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

Polycyclic presentation:

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

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, 363021);