#370027

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	#2487
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321080, #72321148, #72321165, …
maximal normal subgroup of	#72321080, #72321148, #72321165, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277223);