#6379175

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂²×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4622 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)(5,6,10,15,11,14,9,23)(7,16,19,12,17,28,24,26)(8,13,21,32,22,30,25,31)(18,33,35,27,34,29,20,36);(5,7)(6,12)(8,20)(9,24)(10,19)(11,17)(13,29)(14,26)(15,28)(16,23)(18,21)(22,35)(25,34)(27,31)(30,33)(32,36);(2,4)(7,17)(12,26)(16,28)(18,34)(19,24)(20,35)(27,36)(29,33);(5,8)(6,13)(7,18)(9,25)(10,21)(11,22)(12,27)(14,30)(15,32)(16,33)(17,34)(19,35)(20,24)(23,31)(26,36)(28,29);(6,14)(12,26)(13,30)(15,23)(16,28)(27,36)(29,33)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343646, #72343648, #72343699
maximal normal subgroup of	#72343646, #72343648, #72343699

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286371);