#6381905

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60422	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343392, #72343646, #72343683
maximal normal subgroup of	#72343392, #72343646, #72343683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289101);