#356081

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×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	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2326 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1757, #72342934, #72342942
maximal normal subgroup of	D₄×#1757, #72342934, #72342942
maximal quotient of	#72342988

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263277);