#376765

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1275 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283961);