#455949

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1135 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,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, 363145);