#356644

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 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)(5,6,10,15,11,9,14,23)(7,16,19,30,18,12,24,26)(8,13,21,28,22,20,27,33)(17,29,32,36,31,25,34,35);(5,7)(6,12,9,16)(8,17)(10,19)(11,18)(13,25,20,29)(14,24)(15,26,23,30)(21,32)(22,31)(27,34)(28,35,33,36);(2,4)(5,8,10,21,11,22,14,27)(6,13,15,28,9,20,23,33)(7,17,19,32,18,31,24,34)(12,25,26,35,16,29,30,36);(5,9,10,23,11,6,14,15)(7,12,19,26,18,16,24,30)(8,20,21,33,22,13,27,28)(17,25,32,35,31,29,34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,g^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, #72342922, #72342942
maximal normal subgroup of	D₄×#1757, #72342922, #72342942
maximal quotient of	#72342987

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263840);