#356184

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

1421 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)(7,11)(8,18,19,12)(9,14)(10,13)(15,27,28,22)(16,24)(17,23)(20,31,32,25)(21,26)(29,35,36,33)(30,34);(2,3)(5,7,9,16,10,17,21,30)(6,11,14,24,13,23,26,34)(8,15,20,29,19,28,32,36)(12,22,25,33,18,27,31,35);(5,8)(6,12)(7,15)(9,20)(10,19)(11,22)(13,18)(14,25)(16,29)(17,28)(21,32)(23,27)(24,33)(26,31)(30,36)(34,35);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,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, #72342944, #72342945, …
maximal normal subgroup of	D₄×#1757, #72342944, #72342945, …
maximal quotient of	#72342995

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263380);