#449498

order	512
nilpotency class	3
isoclinism	isoclinic class of #417530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3288 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255954, #72256181, #72256257
maximal normal subgroup of	#72255954, #72256181, #72256257
maximal quotient of	#72267306, #72267308

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356694);