#366452

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273648);