#181020

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=i,b^a=be,e^a=eh,c^b=ci,f^b=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88216);