#363500

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1276 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239887, #72239902, #72239911, …
maximal normal subgroup of	#72239887, #72239902, #72239911, …
maximal quotient of	#72239889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270696);