#117764

order	512
nilpotency class	3
isoclinism	isoclinic class of #10018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240294, #72240337, #72240343, …
maximal normal subgroup of	#72240294, #72240337, #72240343, …
maximal quotient of	#72286539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24960);