#210069

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279227, #72309473, #72309513
maximal normal subgroup of	#72279227, #72309473, #72309513
maximal quotient of	#72279385

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117265);