#235680

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	#34936	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213366, #72216051, #72216054
maximal normal subgroup of	#72213366, #72216051, #72216054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142876);