#356355

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

984 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)(7,18,22,31)(8,23)(9,15)(10,16)(11,28)(12,17)(13,21,32,27)(14,24)(19,29,20,33)(25,34)(26,36,35,30);(1,2,3,4)(5,7)(6,13)(8,19)(9,26)(10,27)(11,21)(12,22)(14,29)(15,33)(16,31)(17,32)(18,28)(20,25)(23,36)(24,35)(30,34);(5,8,11,24,12,25,10,9)(6,14,16,23,17,15,28,34)(7,19,21,35,22,20,27,26)(13,29,31,36,32,33,18,30);(5,9,10,25,12,24,11,8)(6,15,16,34,17,14,28,23)(7,20,21,26,22,19,27,35)(13,30,18,33,32,36,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263551);