#151988

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1398, #72319598, #72319605, …
maximal normal subgroup of	D₄×#1398, #72319598, #72319605, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59184);