#368399

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2408
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 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)(7,17)(8,22)(9,25)(10,28)(11,16)(12,20)(13,19)(14,26)(15,29)(18,34)(21,32)(23,30)(24,33)(27,31)(35,36);(1,3,4,2)(5,7,10,20,11,21,29,23)(6,12,15,17,16,30,28,32)(8,18,24,9,19,26,36,27)(13,31,33,14,22,25,35,34);(5,8)(6,13,16,22)(7,18)(9,20)(10,24)(11,19)(12,25,30,31)(14,17,34,32)(15,33,28,35)(21,26)(23,27)(29,36);(5,9,10,26,11,27,29,18)(6,14,15,25,16,34,28,31)(7,19,20,36,21,8,23,24)(12,22,17,35,30,13,32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280888, #72281002, #72281142, …
maximal normal subgroup of	#72280888, #72281002, #72281142, …
maximal quotient of	#72313517

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275595);