#182737

order	512
nilpotency class	3
isoclinism	isoclinic class of #17547
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1108 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89933);