#356189

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1417 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)(5,6,9,15,11,14,10,16)(7,12,18,27,20,26,19,28)(8,21,22,30,23,13,24,29)(17,31,32,36,33,25,34,35);(2,3)(5,7,10,19,11,20,9,18)(6,12,16,28,14,26,15,27)(8,17,22,32,23,33,24,34)(13,25,29,35,21,31,30,36);(5,8)(6,13)(7,17)(9,24)(10,22)(11,23)(12,25)(14,21)(15,30)(16,29)(18,34)(19,32)(20,33)(26,31)(27,36)(28,35);(1,3,4,2)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1757, #72342938, #72342950, …
maximal normal subgroup of	D₄×#1757, #72342938, #72342950, …
maximal quotient of	#72342998

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263385);