#6381915

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	#60450	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2516 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)(3,4)(5,6)(7,15,18,30)(8,12)(9,14)(10,13)(11,17,25,22)(16,26,31,35)(19,32,33,24)(20,28)(21,27)(23,29)(34,36);(5,7,9,22,10,18,23,17)(6,11,14,30,13,25,29,15)(8,19,20,16,21,33,34,31)(12,26,28,24,27,35,36,32);(2,4)(5,6)(7,11)(8,12)(9,14)(10,13)(15,17)(16,24)(18,25)(19,26)(20,28)(21,27)(22,30)(23,29)(31,32)(33,35)(34,36);(5,8,9,20,10,21,23,34)(6,12,14,28,13,27,29,36)(7,16,17,19,18,31,22,33)(11,24,15,26,25,32,30,35);(6,13)(11,25)(12,27)(14,29)(15,30)(24,32)(26,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343602, #72343653, #72343656
maximal normal subgroup of	#72343602, #72343653, #72343656

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289111);