#354905

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×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×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1536 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224727, #72318965, D₄×#1757, …
maximal normal subgroup of	#72224727, #72318965, D₄×#1757, …
maximal quotient of	#72318993, #72343011

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262101);