#185187

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#532	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3642 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92383);