#181611

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1550 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88807);