#354605

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224693, #72226665, #72226671, …
maximal normal subgroup of	#72224693, #72226665, #72226671, …
maximal quotient of	#72313604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261801);