#181516

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1832 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)(6,9)(7,11)(10,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,25)(23,28)(24,27);(5,6)(7,12)(8,10)(9,11)(13,15)(14,19,21,27)(16,24)(17,25)(18,23)(20,22,26,28);(6,10)(9,12)(15,23)(19,27)(22,28)(24,25);(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,22,27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,b^a=be,e^a=eh,g^a=ghi,c^b=ch,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88712);